claims depot shelf
The Twin Prime Conjecture
Formal claims (Lean)
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The central claim is Theorem 1.1. With $z'=\min\{z,x/y\}\to\infty$, $u=\log(z/y)/\log z$, and $\gamma=\log(1/u)/\log\log z'$, the paper proves: if $u\le(\log z')^{-(\gamma_0+\eta)}$ and $y\ge z-z^{17/30+\varepsilon}$ then $x^{-1}H^*(x,y,z)=u(1+o(1))$; if $\eta<\gamma<\gamma_0-\eta$ then $x^{-1}H^*(x,y,z)$ is of order $(\log z')^{-Q((1+\gamma)/\log2)}(\log\log z')^{-1/2}$, with $Q(\rho)=\rho\log(\r
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The central discovery is Theorem 1.1: for any fixed $0.98353<\gamma<1$, one has $\#\{n\le x : n\in P_7 \text{ and } \lfloor n^{1/\gamma}\rfloor \text{ is prime}\}\gg x/(\log x)^2$, with the implied constant depending only on $\gamma$. The prime-factor count is with multiplicity, so $P_7$ includes powers. The proof enlarges the admissible level of distribution $\xi(\gamma)=\min\{(25857\gamma-23041)
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The central discovery is an improved asymptotic answer to the spin geography problem. For even $s\ge 0$, let $\Lambda_s$ be the smallest odd $q$ such that the form $sE_8\oplus qH$ has the $\infty^2$-property, meaning that a simply connected manifold with that intersection form carries infinitely many pairwise nondiffeomorphic irreducible symplectic structures and infinitely many pairwise nondiffeo
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On the paper's own terms, the discovery is a finite reduction of the two-prime-sum problem to a single numerical condition. For $x=12n+4$, the degree in $G(x/2)$ is $n-f(n)+1$ or $n-f(n)$ according as $x-3$ is prime, where $f(n)$ is the size of the union of the divisibility events $A_p$ for primes $3<p<\sqrt{x}$; analogous formulas are proved for $x=12n+k$, $k=0,2,4,6,8,10$. The paper proves $\pro
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The paper's central claim is that every integer $N\ge2$ can be written as $N=a+b$ with $a,b$ positive integers and $\Omega(ab)\le33$. The proof works with the auxiliary sequence $a_n=n(N-n)$, halved when $N$ is odd, and keeps only indices with $3\nmid a_n$; the prime 3 is then excluded from the sifting set. This pre-sieving removes the extremal $w=3$ case from the dimension condition, allowing a s
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On its own terms, the paper establishes that the restarting Josephus survivor map has no hidden period: $\operatorname{Per}(F_n)=L_n\mathbb{Z}$, proved by freezing trajectories and by using the least recurrence level not dividing a proposed shift as a witness. It then proves that $F_n$ is surjective onto $\{1,\ldots,n\}$: a prime-power CRT construction with blocking primes in $(n/2,n]$ produces in
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The central claim is Theorem 1.5: assuming the eventual even difference hypothesis (ED), the set of primes cannot be a spectrum of any Borel probability measure. The proof reduces spectrality to two conditions on $a(k)=|\hat{\mu}(k)|^2$: $a(p-q)=0$ for all distinct primes $p,q$, and $\sum_{p} a(n-p)=1$ for every integer $n$. Under ED, the first condition kills all sufficiently large even arguments
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Corollary 1.2: The arithmetic-progression version is false whenever alpha < 1/2 and beta < 3/8, including the proposed (1/3,1/3), even under n^{o(1)} relaxations.
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Theorem 1.1: If an n-divisor arithmetic progression has height H with log H = o(sqrt n), then its length L is at least (sqrt(8/27) - o(1)) n^(3/4)/sqrt(log n).
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The central claim is Theorem 1.1: for every fixed $R\ge 1$ there are explicitly computable functions $E_1,\ldots,E_R$ in $C^\infty([0,1])$ such that, uniformly for integers $n\ge m$ as $m\to\infty$, $$\log s(n,m) = m\log n - \psi(m) + \log\frac{m}{n} + m\sum_{r=1}^{R}\frac{E_r(m/n)}{(\log m)^r} + o\!\left(\frac{m}{(\log m)^R}\right),$$ where $\psi(m)=\log \operatorname{lcm}(1,\ldots,m)$. The equiv
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The central claim is a two-term asymptotic formula for the Havel–Hakimi residue of the graph $G_n$ on $\{2,\ldots,n\}$ with adjacency $\gcd(x,y)>1$. The paper proves that $$R(G_n)=\bigl(\zeta(2)-1\bigr)\frac{n}{\log n}+\bigl(\zeta(2)-1-A\bigr)\frac{n}{\$log^{2}$ n}+O\!\left(\frac{n}{\$log^{3}$ n}\right),$$ with $A=\sum_{k=2}^{\infty}\log k/(k^2(k-1))=0.3201986326\ldots$. Consequently $R(G_n)\log n
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The central claim is that the three Mertens sums satisfy explicit inequalities with the same shape as the best current Chebyshev-function bounds. In particular, Theorem 2 proves $|\lambda(x)-\log\log x-M|\le A_\lambda(x_0)(\log x)^{1/2}\exp(-C\sqrt{\log x})$ for every $x\ge x_0$, with $A_\lambda(2)=9.2203$ and $C=0.84768$, and also proves $|\lambda(x)-\log\log x-M|\le A_\ell(x_0)/(\log x)^\ell$ fo
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The paper's central discovery is a rigidity statement at the level of the measure \(\Lambda_{f,t} = \sum_p \frac{f(p)}{p}\delta_{\log\log p - t}\): if \(A_1(X_t;f)\to\kappa\) for \(X_t=\exp(e^t)\), then \(\Lambda_{f,t}\to\kappa\, du\) vaguely as \(t\to\infty\). This is shown by writing \(A_1\) and \(A_2\) as convolutions with the Dickman kernels \(K_1(u)=\rho(e^u-1)\) and \(K_2(u)=\$int_1^{{e^u-1}
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The paper's central discovery is a rigidity statement at the level of the measure \(\Lambda_{f,t} = \sum_p \frac{f(p)}{p}\delta_{\log\log p - t}\): if \(A_1(X_t;f)\to\kappa\) for \(X_t=\exp(e^t)\), then \(\Lambda_{f,t}\to\kappa\, du\) vaguely as \(t\to\infty\). This is shown by writing \(A_1\) and \(A_2\) as convolutions with the Dickman kernels \(K_1(u)=\rho(e^u-1)\) and \(K_2(u)=\$int_1^{{e^u-1}
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The core claim is that the least quadratic residue modulo n has exponential order 4^k in the number k of odd prime factors, and that this phenomenon is stable enough to transfer to binary quadratic forms. Theorem 1.1 supplies the two sides: a pigeonhole argument shows ℓ(n) ≤ C k² 4^k, while a finite-field construction, translating primes into irreducible polynomials over F₂[x], produces moduli n f
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The paper proves that W(u) equals 2|Φ_2(u)|cos(arg Φ_2(u)) plus an explicit error θ_2(u), where Φ_2(u)=Φ(u)(1+α(u)), Φ(u)=exp{−uζ+J(−ζ)} divided by the square root of 2πu(1−1/ζ), J(−ζ) is the incomplete gamma function Γ(0,−ζ), and ζ(u) is the unique solution of e^ζ=−uζ in a specified region, equivalently the 1-branch of the Lambert W function applied to 1/u. The error is bounded by 420 u^{−6} for
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The paper's central claim is the exact identity $\operatorname{str}(\det_p)=p$ for every prime p, together with the asymptotic bounds $\operatorname{str}(\det_n)\ge (1-o(1))n^{0.475}$ and $\operatorname{prk}(\det_n)=n-o(n)$. For prime p, the complement of the determinantal hypersurface in the projective space of p×p matrices is $\mathrm{PGL}_p$, and the argument uses the Chow ring computation $\ma
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Theorem 1.4 asserts that as x→∞, the count of n<x with P^+(n)<P^+(n+1) exceeds 0.280x, and by symmetry the same holds for the reverse inequality. The constant comes from a decomposition of n according to the sizes of P^+(n) and P^+(n+1): a cutoff c=0.1348 separates small and large cases, and each piece is estimated either by the standard smooth-number distribution or by a new lower-bound function
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The paper's central claim is that the Skolem Problem reduces to a question about prime gaps. It defines good primes as those that are not 'bad', where badness means that a prime divides the algebraic norm of a certain exponential-polynomial expression associated to a small LRS around a potential large zero. For any large zero of an LRS, the paper shows there must be an interval around the zero con
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For every digit d from 0 to 9, the total number of occurrences of d among all digits of all primes less than 10^n, divided by the total number of such digits, equals 1/10 plus an error of size O((log n)/n) as n tends to infinity, uniformly in d. The same limit therefore holds in total variation for the empirical digit measure.
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For each r > 2 there exist constants r' < c(r) < 2 < C(r) with lim c(r) = 1 and lim C(r) = ∞ such that the r-variation of the prime-weighted Carleson series is bounded on ℓ^p for all c(r) < p < C(r), while the variation is unbounded for p ≤ r'. The same reductions give that the maximal prime Carleson operator is bounded on ℓ^p(ℤ) for every 1 < p < ∞.
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For every fixed k≥2 and every N≥0 the additive Mertens sum admits the complete expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x), where the coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t1,…,log tk)/(t1+…+tk) dt and hn is the complete homogeneous symmetric polynomial of degree n.
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For every fixed k≥2 and every N≥0 the additive Mertens sum admits the complete expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x), where the coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t1,…,log tk)/(t1+…+tk) dt and hn is the complete homogeneous symmetric polynomial of degree n.
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Assuming the Riemann Hypothesis and a mild lower bound on the Gaussian smoothing width relative to a fixed simple critical-line zero, the full Gaussian–Perron prime-force defect near that zero equals the universal selected-zero profile −a Re(e^{−λ}/λ) plus an error that is O(1/log X) plus exponentially small, uniformly on compact sets of the logarithmic displacement λ away from zero.
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For a Serre curve — an elliptic curve over Q whose adelic Galois image is as large as possible, namely the index-2 subgroup forced by the quadratic field Q(√Δ_E) — Theorem 3 provides a closed formula for C^{m-div}_E, the natural density of good primes p with m | #E_p(F_p). Writing m = m_1 m_2 with m_1 = gcd(m, m_E^∞), the density equals the average density C^{m-div} when m_1 is outside a small exc
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The main result, Theorem 2.2, states that if n=pq and n+12=p'q' are both products of two distinct primes with p<q and p'<q', then the two jump conditions φ(n+12)-φ(n)=12 and σ(n+12)-σ(n)=12 hold together if and only if p≥5, q=p+8, and {p',q'}={p+2,p+6}. The proof reduces both equations to the equality p+q=p'+q', then uses the discriminant relation (q-p)^2-(q'-p')^2=48. Factoring 48 gives only thre
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The main result, Theorem 2.2, states that if n=pq and n+12=p'q' are both products of two distinct primes with p<q and p'<q', then the two jump conditions φ(n+12)-φ(n)=12 and σ(n+12)-σ(n)=12 hold together if and only if p≥5, q=p+8, and {p',q'}={p+2,p+6}. The proof reduces both equations to the equality p+q=p'+q', then uses the discriminant relation (q-p)^2-(q'-p')^2=48. Factoring 48 gives only thre
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On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that i
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On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that i
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On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that i
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On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that i
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In the paper's own terms, the discovery is that Chebyshev's bounds follow from comparing two weighted prime sums: π(x) ≍ (∑_{p≤x}√(ln p/p))² − ∑_{p≤x} ln p/p, and (∑_{p≤x}√(ln p/p))² ≍ x/ln x. The second statement is shown by combining Abel's identity with Mertens' estimate; the first is shown by writing the difference E(x) at prime indices and using the exact recurrence E_n = E_{n-1} + 2a_n S_{n-
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The paper establishes an asymptotic formula for the average of the error term appearing in Bateman–Horn’s conjecture, taken over an exponential range of arguments; the resulting main term is strictly larger than square-root size, so the averaged error exhibits worse-than-square-root cancellation.
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On the paper's own terms: Mersenne prime exponents have elevated divisor complexity of p-1 relative to local prime controls, and this elevation survives matching by exponent size. The mean percentile rank of S(p) in a 5000-prime window is 0.648 versus a null expectation of 0.5; sign, Wilcoxon, and permutation tests reject the null with p-values around 0.002. The paper interprets this as evidence f
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On the paper's own terms: Mersenne prime exponents have elevated divisor complexity of p-1 relative to local prime controls, and this elevation survives matching by exponent size. The mean percentile rank of S(p) in a 5000-prime window is 0.648 versus a null expectation of 0.5; sign, Wilcoxon, and permutation tests reject the null with p-values around 0.002. The paper interprets this as evidence f
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On the paper's own terms: Mersenne prime exponents have elevated divisor complexity of p-1 relative to local prime controls, and this elevation survives matching by exponent size. The mean percentile rank of S(p) in a 5000-prime window is 0.648 versus a null expectation of 0.5; sign, Wilcoxon, and permutation tests reject the null with p-values around 0.002. The paper interprets this as evidence f
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Informational cardinality I(M)=(α(M),δ(M),ι(M)) distinguishes the essential fractal prime set P_ess from the generalized Cantor set C_{1/3}: both have continuum cardinality (α=1), yet dim_H(P_ess)=1/2>1/3=dim_H(C_{1/3}) and ι(P_ess)=-ζ(1/2) while ι(C_{1/3})=0, so I(P_ess)>I(C_{1/3}) under lexicographic order. The same framework pairs P_ess with a fractal zero set Z_F of equal dimension and conject
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The central discovery is that a deterministic fractal set, PF, can be built directly from the prime residues modulo 16 and that its fractal dimension is exactly 1/4, with Hausdorff and box-counting dimensions agreeing. At each stage, every interval is split into 16 equal pieces and only two are kept, those indexed by the current prime residue a_n and its opposite a_n+8 mod 16. The proof shows that
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On the author's own terms, the paper establishes that primes have an average distribution in arithmetic progressions compatible with moduli beyond sqrt(x). In particular, Theorem 1.1 asserts that for composite moduli q = q1 q2 with Q1^2 Q2 < x^{1-epsilon} and Q1^7 Q2^12 < x^{4-epsilon}, any divisor-bounded bilinear weight gives a discrepancy sum of size O(x/(log x)^A). Since those two constraints
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Theorems 1.1 and 1.2 assert that for h = X^θ with 2/15+ε < θ < 0.99, all but O_ε(X exp(-c_ε (log X)^{1/4})) integers x ∈ [X,2X] satisfy R(x) ≍ h and S(x) ≍ h, where R(x) counts representations of n ∈ (x,x+h] as p+a (p prime, a in a lacunary set A_λ(X) of size ≍ log X) and S(x) counts distinct such n. The lower bound R(x) ≫ h comes from a quoted almost-all primes-in-short-intervals asymptotic (rest
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On the paper's own terms: a proposition P is α-decidable if there exists a Brouwer ordinal y such that P ↔ y ≥ α. Using the type Brw of Brouwer trees, whose limit constructor accepts only strictly increasing sequences, the paper shows that 1-decidability coincides with ordinary decidability and (ω+1)-decidability with semidecidability. The main theorem (Theorem 7.5) states that for a family P : N
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The central claim is Theorem 1.4: along the sequence of primes p_n, the return-times averages converge pointwise for all second systems, and the Wiener–Wintner averages converge for each fixed frequency θ. The mechanism is to write the prime indicator through the von Mangoldt function, approximate it by a truncated Heath–Brown model, control the U^3 norm of the part of the model with fixed rationa
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The load-bearing result is Theorem 1: for all 2≤n≤4,553,432,387, every row of the Sierpiński matrix S_n contains at least one prime. The proof works because the largest prime gap starting below (4,553,432,387)² is the 82nd maximal gap, of width 1,572, so the pigeonhole principle guarantees at least floor(n/1,572) primes in each row for n in that range. The small cases n≤13 are checked directly, or
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The central claim is that a scaled omega-Gibbs prior on prime multipliers, pi_p proportional to 2^{-beta * ell_omega(p)} with beta > 1, makes the MTE tractable and universal: additive gaps X_{t+1} - X_t have asymptotically Pareto tails with exponent beta, and the time-averaged omega codelength converges almost surely. The pure beta = 1 prior is a critical boundary case: it is regularly varying wit
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Barrett's formula asserts that Barr(n) = 3 + sum_{k=5}^{n-1} floor( sin(pi (k-1)!/k) / sin(pi/k) ) equals the number of primes p < n, under the historical convention that 1 is counted as prime. The summand is a geometric indicator of primality: for composite k ≥ 5, (k-1)! is divisible by k, so the numerator's sine is zero; for prime k, Wilson's theorem forces the ratio to be 1. Thus every integer
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The paper's core discovery is Theorem 8.2: RH holds if and only if the trajectory error functional satisfies E(X) ≪ X^(1/2) log X for all X ≥ e^120. The proof runs through a chain of unconditional estimates—one-visit and parent-window lemmas limiting how often a trajectory hits a window, macro-step alignment showing that L ≍ log X composite steps contract the scale from X to X^(3/4), and a frequen
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Theorem 2.6 of the paper is that the sliding-window 'puzzle' algorithm computes h(k,x) on input x in O(x log x) arithmetic operations and O(x^(2/3)) space. The priority-queue algorithm, which keeps one chain of each length in a min-heap keyed by sum, achieves O(x log x) time and O(x^(3/5)) space when primes are produced by an incremental interval sieve, or O(x(log x)^2) time and O(sqrt(x log x)) s
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The paper's central claim is that old existence theorems for primes in short intervals can be made fully explicit by splicing together three independent effective inputs. Using a cited 90th-power theorem, interpreted by inspection as [x, x + 90 x^(1 - 1/90)] for all real x >= 1, plus the cited cube theorem with its improved threshold, plus the known table of maximal prime gaps and explicit linear
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The central claim is that pointwise ergodic averages along the set of polynomial values of primes of the form x^2 + n y^2 converge almost everywhere for every function in L^p with p > 1, for every measure-preserving system. The proof adapts the Hardy-Littlewood circle method to this arithmetic setting, with the novel ingredient being estimates for the distribution of the prime ideals that correspo
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The central claim is that for an irrational $\alpha$ and fixed $\tau\in(0,1/8)$, the set $\mathcal{P}_\tau(\alpha)=\{p\text{ prime}:\lVert\alpha p\rVert\le p^{-\tau}\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of le
Stated claims
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The central claim is Theorem 1.1: the ordered Goldbach count $G(2n)$ on the antidiagonal $I_{2n}$ is an $\mathcal{O}(n/\ln^{2} n)$ function of $n$. The proof proceeds by induction, partitioning the $(n-2)$ ordered odd pairs that sum to $2n$ into the prime-prime class $G(2n)$, the composite-composite class $D(2n)$, and the mixed classes $H(2n)$ and $K(2n)$, which agree by symmetry of the antidiagonal. From the identity $G(2n)+H(2n)=\pi(2n-3)$ together with the prime number theorem expansion $\pi(x)=x/\ln x+x/\ln^{2}x+2x/\ln^{3}x+\cdots$, the proof fixes the leading and second-order terms of $H(2n-2)$; positivity of $G(2n-2)$ combined with the induction hypothesis forces the second coefficient $\alpha$ to be strictly below $2$. A sixteen-case analysis of the prime and composite configurations of a pair $(x,y)$ and its shift $(x+2,y-2)$ under the maps $\psi\colon I_{2n}\to I_{2n+2}$ and $\tau\colon I_{2n}\to I_{2n-2}$ then yields $H(2n+2)=H(2n-2)+2R_1-2R_3$, where the equalities $R_1=R_2$ and $R_3=R_4$ come from symmetry of the antidiagonals. Whichever of the surviving sizes $R_3$ has, the second coefficient of $G(2n+2)$ is strictly positive, so every sufficiently large even number has a Goldbach representation and the binary conjecture follows for all even integers greater than four; the corollaries give the ternary statement for odd integers and the bounds $D(2n)=O(n)$ and $H(2n)=K(2n)=O(n/\ln n)$.
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Under the condition $\mu < q$, the summatory representation function $A_D(x)$ satisfies $A_D(x) = x^\alpha \Psi_D(\log_b x) + O_D(x^\beta)$ with a unique, strictly positive, H\"older continuous, $1$-periodic profile $\Psi_D$ (Theorem 1.1). The profile admits an explicit Fourier series $\Psi_D(u) = \sum_{k\in\mathbb{Z}} \frac{h_D(k)}{\Gamma(1+\alpha+i\tau_k)} e^{2\pi i k u}$ with $\tau_k = 2\pi k / \log b$, where $h_D(k)$ are the Fourier coefficients of a smooth auxiliary function $H_D$ arising from a periodic factorization of the Laplace generating function (Theorem 1.2). When $D$ is a complete residue system modulo $b$ containing zero, the carry automaton on $K_D = \{-M,\dots,0\}$ admits a zero-synchronizing word if and only if $\gcd(D)=1$ (Theorem 1.3). This synchronization yields an exponentially accurate interval approximation of $S_D$ and a transfer principle for arithmetic weights, leading to the prime number theorem $\pi_D(x) = \frac{x}{\log x} \Psi_D(\log_b x) + O_D\bigl(\frac{x}{(\log x)^2}\bigr)$ (Theorem 1.4) and an $L^1$ scaling limit $\|\mathbf{1}_{E_N} - \mathbf{1}_{E_D}\|_{L^1([0,1])} \leq C_D \rho_D^N$ (Theorem 1.5).
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The central claim is Theorem 1.3: let $h\neq 0$ and let $a,b\in\mathbb{Z}^+$ with $(a,b)=1$, $(b,h)=1$ and $(b,h-1)$ Golomb-prime-free; then for every $y\ge |h|$ there is a finite set $B$ of Golomb primes, all exceeding $y$, with $a/b=\sum_{g\in B} 1/(g-h)$. The two conditions on $b$ are shown in Section 6 to be necessary, not technical: if a prime $p$ divided both $b$ and $h$ then $p$ would have to equal some $g>|h|\ge p$, and if a Golomb prime $q$ divided both $b$ and $h-1$ then $q\mid g-1$, which the defining sieve forbids. The proof rests on two new distribution results: a Siegel--Walfisz-type asymptotic for $|\{g\le x : g\equiv h \pmod b\}|$ that holds for $b\le(\log x)^A$ with the expected Euler-product correction for Golomb primes dividing $b$, and an upper bound saying that among Golomb primes $g\le x$ in the class $h \bmod b$, at most a proportion $0.259+o(1)$ have $g-h$ divisible by a prime power exceeding $x^{0.99}$.
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论文的核心发现包含两个定理。定理1.1：对于任意一族满足常数项为零且次数各不相同的整数多项式$P_1,\dots,P_\ell$，存在$c^*>1$和$\kappa>0$，使得对所有$1<c<c^*$，任意充分大的$N$和任意满足$|A|\geq N/(\log N)^\kappa$的子集$A\subset[N]$，都存在一个Piatetski–Shapiro素数$p=\lfloor n^c\rfloor$，使得配置$x, x+P_1(p-1),\dots,x+P_\ell(p-1)$全部落在$A$中且两两不同。定理1.2：在同样的参数范围内，沿Piatetski–Shapiro素数的多项式多重遍历平均几乎处处收敛，且极限与沿全体素数的平均一致。
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The central claim is that the orbit of any positive real number under the free semigroup generated by the affine maps T_0(x)=x/a and T_1(x)=bx+1 (a,b>1) is exponentially dense: for every x>0 and u≥0 there exist infinitely many words γ with |γ(x)−u| < C $λ^{{−|γ|}}$, where λ = exp( (log a · log(a/(a-1))) / (log($a^{2}$ b/(a-1))) ) > 1 and C depends on a,b,x,u but not on γ. This improves the previously known denseness result of Bergelson, Misiurewicz and Senti from a qualitative to a quantitative exponential rate. For the Möbius pair, the paper establishes a complete quantitative picture: O($n^{{−1}}$) denseness for all targets, a rate $n^{{−2}}$ for irrational targets, exponential rates for targets with bounded partial quotients, and optimality of the $n^{{−1}}$ rate for rational targets from rational starting points.
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The central claim, Theorem 1, asserts that for every noninteger $c>1$, every $1<p<\infty$ and every $r>2$, on every $\sigma$-finite measure-preserving system $(X,\mathcal{B},\nu,T)$ and every $f\in L^p(X)$, $$\bigl\|\mathcal{V}^r(A_N^c f:N\in\mathbb{N})\bigr\|_{L^p(X)}\lesssim_{c,p,r}\|f\|_{L^p(X)}.$$ Because the homogeneous $r$-variation seminorm controls the oscillation of the whole sequence of averages, the estimate yields almost-everywhere and $L^p$ convergence of $A_N^c f$, and Corollary 2 transfers the conclusion to the prime-normalized averages $P_N^c f(x)=\frac{1}{\pi(N)}\sum_{p\le N}f(T^{\lfloor p^c\rfloor}x)$, which share the same limit. Theorem 3 is the Möbius analogue: $\frac{(\log N)^A}{N}\sum_{n\le N}\mu(n)f(T^{\lfloor n^c\rfloor}x)\to 0$ almost everywhere for every $A>1$, every $1<p<\infty$ and every $f\in L^p(X)$. The load-bearing discovery behind all three statements is Proposition 4.2, $$\sup_{\xi\in\mathbb{T}}\left|\frac{1}{N}\sum_{n\le N}(\Lambda(n)-1)e(\xi\lfloor n^c\rfloor)\right|\lesssim_{c,A}(\log N)^{-A}\quad(A>0),$$ which says that at every frequency simultaneously the prime-weight discrepancy $\Lambda-1$ decorrelates from fractional-power phases with arbitrary logarithmic savings, forcing the weighted averages to track the unweighted averages $B_N^c f$, whose variation is controlled separately in Proposition 3.4.
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The paper claims that the function family $F(t_1,\dots,t_k) = Q(1+\varepsilon - P_1) / \prod_{i=1}^k (1+at_i)$, with $P_1 = \sum t_i$, $a \approx k\log k$, and a polynomial $Q$ of degree up to 20, provides a highly efficient basis for the Maynard–Tao sieve. Using the convolution structure of the key integrals $I_\varepsilon(F)$ and $J_\varepsilon(F)$, the author reduces the $k$-dimensional optimization to a single-variable Laplace transform, then computes the resulting Rayleigh quotient numerically with rigorous error bounds. The central numerical result is that $M_{k,\varepsilon} > 4m$ is achieved for $k$ as small as 3438 when $m=2$, leading to the claimed $H_m$ bounds. In the asymptotic regime, the paper explicitly evaluates the leading terms of $M_k$ and obtains a constant $c = 0.273572$ that measures the gap to the Polymath8b upper bound.
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The paper establishes three theorems. Theorem 1: For each of the nine class‑1 imaginary quadratic fields, there exist admissible sets of size 50 (and 49, 48) contained in short intervals of rational integers, yielding new records on the smallest rational integer difference between large primes in those fields (e.g., ≤136 for Gaussian primes, ≤93 for Eisenstein primes). Theorem 2: When admissible sets are allowed to be drawn from two‑dimensional discs, the smallest norm difference between distinct prime elements in the field is reduced; for Gaussian primes the bound is ≤218, and similar improvements are given for the other eight fields. Theorem 3: The generalization of the second Hardy‑Littlewood conjecture to these fields — that the number of primes in a disc is maximized when the disc is centered at the origin — is false for all except Q(√-11); explicit sets of prime elements are provided as counterexamples.
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The core discovery is that the bipartite network R(N) – whose nodes are all integers from2 to N except primes larger than N/2, with edges between a prime and any composite having that prime in its factorization – has a diameter of exactly 5 for every positive integer N. The proof constructs a path of at most five steps between any two nodes and exhibits a concrete pair (an odd composite and a prime from (N/3,N/2]) whose shortest path is exactly five steps. The existence of such a prime (Lemma 1) is established using Rosser–Schoenfeld bounds on the prime-counting function and a manual check for small N, so the diameter is unconditional and does not depend on open conjectures. Additionally, the subnetwork R'(N) (which omits primes larger than √N) has diameter 4, confirming that both networks are small-world. The paper also proves that boundary nodes are exactly prime powers and primes in (N/3,N/2], and shows that the classic disconnection algorithm (Algorithm 1) misidentifies many low-degree primes as relevant; two alternative algorithms (weighted-degree-difference and no-disconnection) give a much smaller and more plausible set of relevant nodes (essentially the primes ≤ √N). Finally, the Shannon and Tsallis information dimensions are computed numerically: both decrease with N, and for box sizes ε =3 or 4 the number of covering boxes becomes roughly constant.
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On its own terms, the paper's central claim is Theorem 1.1: under Hypothesis 2.1 — the uniform two-over-two zeta-ratios formula (2.3) with error $O_\varepsilon(U^{1/2+\varepsilon})$ for shifts with real parts in $[3/64,5/64]$ and imaginary parts of size at most $U^{1/4}$ — one obtains $\sum_{n\le x}\Lambda(n)\Lambda(n+4)=2C_2x+o(x)$, the same asymptotic for $A_4(x)$, and $\pi_4(x)\sim 2C_2x/(\log x)^2$, where $C_2$ is the twin-prime constant and $S(4)=2C_2$ is the singular series for difference four. The quantitative engine is Theorem 1.2: for a fixed real smooth weight $W$ supported in $(1,101/100)$, the correlation sum with weight $W(n/x)^2/n$ equals $2C_2\int_0^\infty W(u)^2\,du/u$ with error $O_{W,\varepsilon}(x^{-1/16+\varepsilon})$, improved to $O_{W,\varepsilon}(x^{-3/8+\varepsilon})$ if the Riemann hypothesis is added. Two corollaries extend the reach: the logarithmically weighted prime-pair sum satisfies $B_4(x)=2C_2\log x+o(\log x)$, and the second finite difference of integrals of squared increments of the truncated Chebyshev function equals $4C_2x+o(x)$, tying the pair problem to a finite Nyman-Beurling approximant. Throughout, the zero-free strip (2.6), the zero-count bound (2.8), and the differentiated ratio mean (2.14) are derived from the hypothesis itself rather than assumed.
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Theorem 1.1: For every primitive quadratic Dirichlet character $\chi$ of conductor $q \ge 3$, any real zero $\beta \in (0,1)$ of $L(s,\chi)$ satisfies $(1-\beta)\log q > 1/100$. The proof proceeds in three main parts. The arithmetic input (Proposition 3.1) gives an explicit upper bound for $S_-(x,\chi) = \sum_{p\le x,\, \chi(p)=-1} (\log p)/p$, obtained via an injective evaluation map on integer-coefficient polynomials and a triangular basis that supplies prime-power divisibility. The real-variable part (Proposition 4.1) estimates the sum $H_\beta(t) = \sum_{n\le t} a(n)/n^\beta$ at a real zero, with an error term that avoids a factor $1/(1-\beta)$ by pairing the tails of two series. Finally, Lemma 5.1 converts these estimates into a logarithmically weighted prime-sum comparison; assuming $(1-\beta)\log q \le 1/100$ leads to an explicit numerical contradiction. The same argument with adjusted exponents (Appendix A) yields the stronger constant $1/64$. The entire proof uses only elementary methods and a single set of parameters for all conductors.
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On the assumption that the Riemann hypothesis is true, the note claims three explicit statements about where primes must lie. Theorem 1: for every $x \geq 3.5\times 10^{90}$ the interval $[x, x+x^{1/2+1/40}]$ contains a prime — the 2001 Baker–Harman–Pintz theorem with its 'sufficiently large' threshold filled in. Theorem 2: for every $x \geq 1$ and every $0 < m \leq 15.831$, the interval $[x, x+x^{1/2+1/m}]$ contains a prime. Theorem 3: for every $y \geq 1$ and every $M \geq 2.29$ there is a prime between $y^M$ and $(y+1)^M$. The small-$x$ half of Theorem 2 is delegated to a computer verification of the upper Oppermann conjecture (a prime between $n^2$ and $n^2+n$) through $n = 7\times 10^{13}$, i.e. $x \leq 4.9\times 10^{27}$; the large-$x$ half is a direct comparison of interval sizes, with crossover at $x_* = \exp(-40\,W_{-1}(-5/176)) \approx 3.4828\times 10^{90}$ for $m=40$ and the exponent bound $m < \ln x_{**}/(\ln\ln x_{**} + \ln(22/25)) = 15.83179525$ at $x_{**} = 4.9\times 10^{27}$. Theorem 3 follows from Theorem 2 through the substitution $x = y^M$ with $M = 2m/(m-2)$, which rewrites the target interval as $[y^M, y^M + y^{M-1}]$, contained in $[y^M, (y+1)^M]$.
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The paper proves that for any real quadratic function field K over F_q(T) (q odd) and any pair (x_1, x_2) in k_∞^2 not lying in the diagonal embedding of K, there exist infinitely many principal prime ideals p of O_K and elements b ∈ O_K such that |x_i - σ_i(b)/σ_i(π)| ≤ N(p)^{-2/3+ε} for i=1,2, where π is a suitably normalised generator of p. This is the exact analogue of Matomäki's classical exponent θ=1/3. The result is unconditional because the Riemann hypothesis for Hecke L-functions holds for function fields (Weil). The proof overcomes the obstacle of infinitely many units in the real quadratic case by using Hecke characters of non-trivial infinity type, whose infinity types form a finite set (parametrised by n modulo the regulator R). Consequently, the Fourier expansion of a Dirichlet character on normalised prime generators becomes an exact finite linear combination of Hecke character values, avoiding the truncation errors that occur in the number-field analogue.
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Sampling at Gram points is far from neutral: the full cloud $\mathcal{C}_N=\{E(g_n)\}$ and the prime cloud $\mathcal{P}_N=\{E_1(g_n)\}$ form highly structured subsets of the ambient curves traced by $E(t)$ and $E_1(t)$ as $t$ varies. In the full cloud the good/bad separation is forced by the approximate functional equation, which gives $Z(g_n)=2(-1)^n\Re E(g_n)+O(g_n^{-1/4})$; in the prime cloud goodness is not defined, yet the isolated bad points still organize into two lobes above and below the good-point cloud. Folding the two lobes onto each other reveals a principal direction along which the viscosity $\mu(g_n)=|Z'(g_n)/Z(g_n)|$ varies systematically: the smallest viscosities cluster at one extremal end, near the conjectural repulsion boundary $\mu=4$, and the isolated-bad viscosity distribution has a sharp lower edge just above that value, with an observed minimum of about 4.26178. The lobe dichotomy is itself geometric: the upper lobe corresponds to the confined zero pair lying to the right of $g_n$, the lower lobe to the pair lying to the left, and the extremal low-viscosity points correspond to confined pairs in which one zero is displaced toward the outer wall $g_{n\pm1}$ of the local Gram interval. The data further suggest the normalized asymptotic lower limit $\liminf (2\pi/\log(g_n/2\pi))\,\mu(g_n)=2$ for isolated bad confined points. All of this happens at modest height, with $N(g_n)\le 164$ so the prime layer has at most 38 terms, a caution the paper itself flags against over-reading the rank correlations. The Davenport–Heilbronn analogue exhibits neither the viscosity bound, since its isolated-bad distribution has substantial mass below $\mu=4$ with a representative value of about 1.45, nor the two-lobe splitting, indicating that both phenomena are tied to the zero-protecting dynamics of the Hardy $Z$-function rather than to the formal functional equation alone.
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The central claim is Theorem 1: as $x\to\infty$, $\sum_{p\le x}\mu(p+2)=o(\pi(x))$. Since $\mu$ is $\pm1$ on squarefree integers and $0$ otherwise, the theorem says the $+1$ and $-1$ contributions among shifted primes cancel to lower order than the total number of primes. The paper then proves the companion squarefree density $\sum_{n\le x}\Lambda(n)\mu^2(n+2)=A_2x+O(x/\log^A x)$ for every fixed $A$, with $A_2=\prod_{p>2}(1-1/(p(p-1)))>0$, and combines the two to obtain Theorem 2: for each $\kappa\in\{-1,1\}$, $\#\{p\le x:\mu(p+2)=\kappa\}=(A_2/2+o(1))\pi(x)$. In particular, among primes $p$ with $p+2$ squarefree, the values $1$ and $-1$ each occur with asymptotic density $1/2$.
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On the paper's own terms, the discovery is a chain of four conditional theorems. Assuming QRH 7/8 — that all nontrivial zeros $\rho$ of the Riemann zeta function satisfy $\operatorname{Re}(\rho) \le 7/8$ — the paper establishes, for all $n \in \mathbb{Z}^+$: $\pi((n+1)^8) > \pi(n^8)$, $\pi(n^8 + 8n^7) > \pi(n^8)$, and $\pi(n^8 + n^7) > \pi(n^8)$, and for all real $x > 1$: $\pi(x + x^{7/8}) > \pi(x)$. Each inequality states that at least one prime exists inside an explicit window; the narrowest integer window has relative length $1/n$, and the real window has relative length $x^{-1/8}$, so all four windows shrink to a vanishing fraction of the location as the numbers grow. The derivation is completely explicit: the hypothesis enters only through a companion preprint's two-sided estimate on the second Chebyshev function $\psi(x)$, which the paper converts into asymmetric bounds on the first Chebyshev function $\vartheta(x)$ and evaluates at the endpoints of each window, with the finite leftover ranges disposed of by direct computation. The paper states plainly that none of this stands without the hypothesis, which it describes as claimed by an unreleased AI-generated preprint collection rather than proved.
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Theorem 1.1: For any fixed prime p, there exist constants γ1, C1, c1 >0 such that for q=p^γ with γ≥γ1 and any q_• ≤ q satisfying log q_• ≥ C1 (log q)^{2/3} (log log q)^{4/3}, the weighte sum over primes ℓ1,ℓ2≤q, ℓ3≤q_• with product ≡ a (mod q) equals ($q^{2}$ q_•/φ(q))(1 + O(exp(−c1 log q_•/((log q)^{2/3}(log log q)^{1/3})))). Hence for sufficiently large γ, every a modulo q is a product ℓ1ℓ2ℓ3 with ℓ1,ℓ2≤q and ℓ3 ≤ exp(C1(log q)^{2/3}(log log q)^{4/3}) = $q^{{o(1)}}$. Theorem 1.2: The number of reduced classes not in EOS2(q) is at most φ(q) exp(−C2(log q)^{1/3}(log log q)^{-1/3}).
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The central claim, stated as Theorem 1, is that for $\frac12<θ<\frac47$ the mean-value estimate (1) holds whenever $R(s)$ is prime-factored and the exponents satisfy $|α₁-α₂|<2θ-1$ and $1-α₁-α₂ < γ(θ)$, where $γ(θ)$ is the maximum of three families of explicit piecewise-linear functions $γ₁_{g₁}(θ)$, $γ₂_{g₂}(θ)$ (for $g₂\ge5$ with $2θ-1$ in a specific interval $I_{g₂}$), and $γ₃_{g₃}(θ)$ (with $g₃$ depending on $θ$). The term $γ₁$ reproduces the 2001 Baker–Harman–Pintz bound; the new terms $γ₂$ and $γ₃$ enlarge the admissible region. At $θ=0.52$, $γ$ increases from $0.08$ to $\frac{19}{225}-ε≈0.0844$, a $5.5\%$ improvement. Theorem 2 provides additional regimes (i)–(iii) covering all $\frac12<θ<1$, and Theorem 3 asserts the existence of primes in $[x-x^{0.52-c},x]$ for some $c>0$.
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The paper's central contribution is twofold. First, it makes explicit the argument (due to Pollack and Pomerance) that Hypothesis A+—the existence of $y^{{1-o(1)}}$ primes p≤5y with p+1 free of prime factors exceeding y^δ for every δ>0—implies h(x)>$x^{{2-ε}}$ for every ε>0, with the pairs squarefree. This implication is written out with explicit constants and formalized in Lean 4, providing an unconditional machine-checked proof of the conditional statement. Second, the paper checks that the OpenAI preprint's construction of smooth shifted primes for p−1 extends to p+1 by a sign change in two sections. On that basis, it states that if the preprint's core estimates (about 2800 lines, unverified) are correct, then Hypothesis A+ holds and consequently h(x)=$x^{{2-o(1)}}$. The authors do not vouch for the preprint and emphasize that its main new method remains unverified.
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The paper claims to have proven that the exponential sum of the Möbius function with linear phases satisfies |∑_{x<n≤x+H} μ(n)e(nα)| ≪ H (log x)^{-1/3+ε} for H ≥ $x^{{0.58+ε}}$, uniformly in α, and that the analogous sum for the von Mangoldt function satisfies |∑_{x<n≤x+H} Λ(n)e(nα)| ≪ H (log x)^{-A} for any A unless ∥qα∥ is very small for some small q. These results improve the previous admissible exponent from 3/5 to 0.58, and the method shows that any further refinement of large value estimates for Dirichlet polynomials will lower the exponent further.
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The central claim is the prime number theorem for terminal blocks (Theorem 1.6): under the algebraic hypotheses that the dominant root $\varphi$ is a Pisot number and the characteristic polynomial is its minimal polynomial, for every fixed concatenation $B$ of semi-proper blocks the count of primes $p \le X$ whose first $e$ terminal blocks form $B$ is asymptotic to $\pi(X)/\varphi^{\mathrm{len}(B)}$. The density is not fitted: the terminal blocks behave asymptotically like independent draws from the probability law $\nu(b_{i,j}) = \varphi^{-i}$, whose masses sum to one by the Pisot identity $\sum_i h_i \varphi^{-i} = 1$. The same toral model proves the companion prime-genericity theorem for the substitution fixed word, namely that every finite factor occurs at prime starting positions with its ordinary factor frequency, which is the Tribonacci statement suggested in earlier work. The discovery underlying both theorems is that terminal-block membership, although not a congruence, is a linear condition: the expanded fixed word lists the terminal blocks of all integers in natural order, stable evaluation at the non-dominant conjugates sends each terminal class to a Rauzy window of relative volume $\varphi^{-\mathrm{len}(B)}$, and the congruence $\mathrm{eval}_\Gamma(n) \equiv n s_0 \pmod{T}$ turns the whole system into an orbit on a compact torus.
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The paper's central claim is Theorem 1.1: whenever $X^{5/8+\varepsilon}\le H\le X$ and $2\le w\le \exp((\log X)^{1/10})$, one has $\|\Lambda-\tilde\Lambda_w\|_{U^k(X,X+H]}\ll_{k,\varepsilon} w^{-a_k}+\exp(-(\log X)^c)$ with $a_k=2^{-k-2}$ and $c=c(k,\varepsilon)>0$, where $\tilde\Lambda_w$ is the Siegel-corrected Cram\'er model at level $w$. This is the first quantitative Gowers bound for the primes in short intervals; it strengthens the long-interval quantitative bound (1.1) and converts the qualitative short-interval bound (1.2) into an effective one, with quasipolynomial savings in the roughness parameter $w$. The author frames the result as a step toward a larger program of quantitative uniformity for arithmetic functions, and reports an application, joint with Richter, to sum-product patterns in the shifted primes.
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The central claim is Theorem 1.6: a weighted count of triples $(d,m,n)$, evaluating the von Mangoldt function at $k+2$ affine forms, is $\gg \tau(D)\varphi(W_1)MN/W_1$ for all large $N$. Since each factor forces a prime value, this lower bound produces many $d,m,n$ with $\Delta dm + 1$, $\Delta^2Dmn + 1$, and $L_j(\Delta dm, \Delta D n/d) + 1$ all prime, and re-labelling $x = \Delta dm$, $y = \Delta D n/d$ yields the patterns $\{x, xy, L_1(x,y), \ldots, L_k(x,y)\}$ inside $\mathbb{P}-1$. From here the paper derives Theorem 1.3 (arbitrary non-proportional linear forms $L_j = a_j x + b_j y$) and Corollary 1.4 ($\{x, x+y, \ldots, x+ky, xy\}$). The authors also state the boundary of their result: finding $\{x,y,xy\}$, let alone $\{x,y,x+y,xy\}$, in $\mathbb{P}-1$ for arbitrarily large $x,y$ would settle a major open problem on the Liouville function, so the pattern's form is essentially the strongest possible.
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The central claim is Theorem 1.1. Fix $\Theta \in (3/4, 1]$ and a choice of the parameter $A$ in Granville's model; for each modulus $q \geq 16$ let $X_q$ be the largest power of two with $2X_q \leq q^{\Theta}$. Then, almost surely, there is a number $q_0$ such that for every modulus $q \geq q_0$ and every reduced class $a \in (\mathbb{Z}/q\mathbb{Z})^{\times}$, there exist distinct $g_1, g_2 \in G \cap (X_q, 2X_q]$ with $g_1 g_2 \equiv a \pmod{q}$. Moreover, writing $N_q(a)$ for the number of such ordered pairs, one has almost surely $\max_a \bigl| N_q(a)\cdot q\log^2 X_q /(c_q X_q^2) - 1\bigr| \to 0$, with $c_q = \frac{\varphi(q)}{q}\prod_{p \leq A(X_q),\, p \mid q}(1 - p^{-1})^{-2}$ the natural local correction. Since $(X_q, 2X_q] \subseteq [1, q^{\Theta}]$, this is the analogue of Erdős–Odlyzko–Sárközy in the wider range $q^{\Theta}$, for all moduli, and the exponent is sharp for the method: the error $O(q^{1/2+o(1)})$ from the lattice-point count balances the main term $X^2/q$ when $X = q^{3/4}$, and the statement certainly fails below $\Theta = 1/2$.
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The central claim, stated as Theorem 1.1, is that for any positive rational $a/b$ in lowest terms with $b$ squarefree there are distinct integers $1<n_1<\cdots<n_k$, each a product of two distinct primes, with $a/b=\sum_i 1/n_i$. The argument first reduces to small targets $\tau=a/b\le 1/400$, then builds one complete bipartite graph: the small side $V=\{2\}\cup B\cup W$, where $B$ holds the primes of $b$ and $W$ the primes in $(y^2,2y^2]$, and the large side $U$ the primes in $(y^8,Y]$ with $Y$ tuned so that half the total reciprocal mass of the graph sits just below $\tau$. Any subgraph then has mass at most $1/2$, so if its reciprocal sum is congruent to $\tau$ modulo 1 it is exactly $\tau$. Counting subgraphs as a finite Fourier sum, the frequencies split into three cases governed by a $V\times U$ table: the small integer frequencies give a positive main term of size $1/L$, and the minor arcs contribute at most $1/(2L)$ by a divisor-counting separation lemma and a no-wrap-around form of the Chinese remainder theorem. Because $y$ can be taken arbitrarily large, several copies of one small target can be padded into disjoint prime intervals, so $q$ copies of $a/(bq)$ sum to $a/b$.
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The central claim is Theorem 1.2: assume infinitely many Siegel zeros of strength 7, meaning real zeros of $L(s,\chi_j)$ at $1-\beta_j = o((\log D_j)^{-7})$ with escalating conductors $D_j$. Then for every irrational $\alpha$ of finite type, every real $\beta$, and every $0<\eta<\eta_0=(\sqrt{e}-1)/(3(1+2\sqrt{e}))=0.0503183978\ldots$, there are infinitely many primes $p$ with $\|\alpha p+\beta\|\le p^{-1/3-\eta}$. The companion Theorem 1.1, under merely infinitely many Siegel zeros, gives the same with exponent $1/3+\varepsilon$. The argument proceeds through an asymptotic formula at the one-third scale (Theorem 3.1) and a lower bound past it (Theorem 4.1), each derived from the exceptional-character identity and sieve, with the improvement quantified by the constant $c(\gamma)=1-2\log((1+3\gamma)/(1-6\gamma))$.
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The central claim is Theorem 1.1: for all sufficiently large $x$, $Y(x) \gg x\log x/((\log_2 x)\log_3 x)$, where $Y(x)$ is the length of the longest interval coverable by choosing one residue class modulo each prime $p \le x$. By the Chinese remainder theorem such a covering produces a run of consecutive composite numbers, so the bound transfers to prime gaps: $G(X) \gg (\log X)(\log_2 X)/((\log_3 X)\log_4 X)$. The decisive step is a bias in the preliminary residue choices, the power tilt $\lambda_p = p^{-\tau}$ with $\tau = t/\log x$, which gives every squarefree $w$-rough composite candidate the survival probability $A n^{-\tau}$ with $A$ of Mertens scale. Joint survival of candidate groups is controlled by the radical of their product, with an error made negligible by the huge choice $w = (\log x)^{100}$; correlations between distinct groups contribute only the factor $G^{\tau}$ from shared primes, which averages to $1 + O((\log x)^{-90})$. Tuning $t = 2\log_3 x + 2\log_4 x$ makes the expected number of uncovered composites $o(x/\log x)$, while the prime survivors stay below the supply of final reserve primes, so the covering is completed by mopping up. The gain over Rankin, the factor $(\log_2 x)/(\log_3 x)^2$, already appears before any method for covering primes in groups is invoked.
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The central claim is Theorem 1.1: for any ε > 0, there exist infinitely many integers n such that the least integer K with d((n+K)!) ≥ 2 d(n!) satisfies K(n) > (3/(4π²) − ε)(log n)(log₂ n)(log₄ n)/(log₃ n)². This improves the previous lower bound (1/9)(log n)(log₂ n)(log₄ n)/(log₃ n)³ established in earlier work, by a factor of order log₃ n, and matches the order of a classical lower bound for gaps between consecutive primes.
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Theorem 1.4 asserts that for any 3 ≤ y << $x^{{o(1)}}$, #{p ≤ x : P_y^+(p-1) < P_y^+(p+1)} ∼ π(x)/2. Theorem 1.6 extends this to y = x^α with 0 < α < 17/32, giving #{p ≤ x : P_y^+(p-1) < P_y^+(p+1)} ≥ (h(α)+o(1))π(x) for some h(α) > 0 that tends to 1/2 as α → 0+. The author also proves an analogue for consecutive integers (Theorem 1.5) and shows that both bounds and the lower bound are optimal up to the stated ranges.
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Theorem 3.1 of the paper states that for any function f from primes to the natural numbers, if f(p)+pZ represents the Montgomery reduction REDC_{p,R(p)} (i.e., f(p) is the modular inverse of R(p) modulo p) for all but finitely many primes p, then the image of f in the ring A = (∏_p F_p)/(⊕_p F_p) is naively transcendental over Q. The proof reduces the problem to the case where f is the inverse of R(p), and shows that R(p) — a power of a fixed base B bracketing p — defines a transcendental element in A by Proposition 2.2, because the set S = {B^e} is sparse enough to satisfy the condition x log e_S(x) log e_S(x−k) ∈ o(e_S(x−k)). The transcendence of the inverse follows because the product of R(p) and its inverse is 1 in each component, and if the inverse were algebraic then R(p) would also be algebraic, a contradiction.
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The central discovery is twofold. First, given coprime u,v, there are infinitely many primes ℓ ≡ u mod v with the Diopantine approximation exponent 1/4 (up to a logarithm), generalising the unconditional result of Vaughan from all primes to a fixed congruence class. Second, an explicit uncountable family of irrationals is produced whose convergents all have denominators that are prime and congruent to u mod v, yielding infinitely many primes ℓ with ||γℓ|| < $ℓ^{{-1}}$, and a criterion based on Xylouris's bound on the least prime in an arithmetic progression shows that this family contains non‑Liouville numbers.
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The central claim is that, under the digit non-resonance condition (every prime divisor of $\gcd\{a-a': a,a'\in\mathcal{D}\}$ divides the base $b$) and $\alpha_{b,D}$-admissibility of the restricted-digit Fourier transform $\hat{C}_k$, the von Mangoldt-weighted counts have the exact shapes $T_k^{(2)} = \bigl(1 - \tfrac{1}{(b-1)|\mathcal{D}|}\sum_{a\in\mathcal{D}} a\bigr)\, S_{b,D}^{(2)}\, X|\mathcal{C}_k| + O_{b,D,A}\bigl(|\mathcal{C}_k|X(\log X)^{-A}\bigr)$ and $T_k^{(3)} = \kappa_{b,D}\, S_{b,D}^{(3)}\, X|\mathcal{C}_k| + O_{b,D,A}\bigl(|\mathcal{C}_k|X(\log X)^{-A}\bigr)$, where $S_{b,D}^{(2)}$ and $S_{b,D}^{(3)}$ are finite averages of local congruence factors over the least significant digit and $\kappa_{b,D}$ is the limiting average of the archimedean weight $(1-2m/b^j)_+$ over the digit set. The leading constants are positive exactly when $b$ is odd or $\mathcal{D}$ contains an even digit (pairs) and when $\min\mathcal{D} < (b-1)/2$ together with $\mathcal{D}\cap\gcd(b,6)\mathbb{Z}\neq\emptyset$ (triples). In the positive cases, after removing proper prime powers and the diagonal, genuine primes realize $\gg X|\mathcal{C}_k|(\log X)^{-2}$ pairs and $\gg X|\mathcal{C}_k|(\log X)^{-3}$ three-term progressions with restricted-digit (common) differences. The new input for triples is a three-point minor-arc lemma: for a minor outer frequency $\alpha$, at least one of the three prime frequencies $\beta$, $\alpha-2\beta$, $\beta-\alpha$ in the kernel $\hat{H}_\Lambda(\alpha) = N^{-1}\sum_\beta \hat\Lambda_X(\beta)\hat\Lambda_X(\alpha-2\beta)\hat\Lambda_X(\beta-\alpha)$ decays like $(X^{-1/5}+(DB)^{-1/3+\epsilon})X(\log X)^4$, which forces the stricter threshold $1/6$.
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The paper's central claim is the numerical bound $H_1 \le 240$ for the liminf of gaps between consecutive primes. According to the abstract, the route is a combination of the Bombieri–Vinogradov theorem with newer equidistribution estimates for smooth moduli; the paper asserts that these two inputs fit together cleanly enough to certify the smaller diameter. If the claim is right, the least possible gap between consecutive primes is now known to be no larger than 240, and the previous record of 246 is improved.
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The paper establishes that the average over the family F of elliptic curves of π'_E(x;k,n) — the count of primes p ≤ x with p ≡ k mod n for which the reduction E(F_p) has prime order — equals C'_{n,k} x / $log^{2}$ x + O(x / (φ(n) $log^{3}$ x)), where C'_{n,k} is the explicit Euler product given in equation (2). This holds for any coprime 1 ≤ k ≤ n, any fixed C, ε > 0, provided A,B > x^ε and AB > x $log^{{10}}$ x, and n ≪ (log x)^C. The constant C'_{n,k} matches the average of the conjectural constants C'_{E,n,k} computed by Lee, Mayle, and Wang, demonstrating that the arithmetic progression analogue of Koblitz's conjecture holds on average.
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The central claim is Theorem 1.1. With $z'=\min\{z,x/y\}\to\infty$, $u=\log(z/y)/\log z$, and $\gamma=\log(1/u)/\log\log z'$, the paper proves: if $u\le(\log z')^{-(\gamma_0+\eta)}$ and $y\ge z-z^{17/30+\varepsilon}$ then $x^{-1}H^*(x,y,z)=u(1+o(1))$; if $\eta<\gamma<\gamma_0-\eta$ then $x^{-1}H^*(x,y,z)$ is of order $(\log z')^{-Q((1+\gamma)/\log2)}(\log\log z')^{-1/2}$, with $Q(\rho)=\rho\log(\rho/e)+1$. At the first transition $\gamma\approx\gamma_0=\log4-1$ the two formulae join, the order below the transition carrying a factor $((\gamma_0-\gamma)\sqrt{\log\log z'}+1)^{-1}$. For $0\le\gamma\le0.01$ the order is known only up to a factor $V$ that is bounded except when $\gamma\to0$ and $\lambda_0\to\infty$ simultaneously, and the authors conjecture that in that exceptional regime the sharp factor is $\gamma+(\lambda_0+1)^{-1/2}$. Ford's earlier result for $z=x$ is recovered from the same statement.
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The central discovery is Theorem 1.1: for any fixed $0.98353<\gamma<1$, one has $\#\{n\le x : n\in P_7 \text{ and } \lfloor n^{1/\gamma}\rfloor \text{ is prime}\}\gg x/(\log x)^2$, with the implied constant depending only on $\gamma$. The prime-factor count is with multiplicity, so $P_7$ includes powers. The proof enlarges the admissible level of distribution $\xi(\gamma)=\min\{(25857\gamma-23041)/(18432\gamma),(357\gamma-314)/(282\gamma)\}-\eta$ in the mean value estimate (3.2), then uses that $\xi$ in a weighted sieve to show the contribution of $P_8$ terms is strictly smaller than the weighted sifting sum, leaving a positive lower bound carried by $P_7$ terms.
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The central discovery is an improved asymptotic answer to the spin geography problem. For even $s\ge 0$, let $\Lambda_s$ be the smallest odd $q$ such that the form $sE_8\oplus qH$ has the $\infty^2$-property, meaning that a simply connected manifold with that intersection form carries infinitely many pairwise nondiffeomorphic irreducible symplectic structures and infinitely many pairwise nondiffeomorphic irreducible nonsymplectic structures. The paper constructs a new family $W_n$ of simply connected spin symplectic 4-manifolds, one for each prime $p\ge 5$, with signature $\sigma(W_{12p}) = 9216p^5 - 9216p^3 - 4608p^2 + 32p + 64$ and $c_1^2(W_{12p}) = 82944p^5 + 36864p^3 - 16128p^2 - 960p + 304$, so their Chern slope $c_1^2/e$ tends to $3$ from below. Applying a spin symplectic geography theorem to these manifolds yields $\Lambda_s \le -10s - 1 + \frac14 c_1^2(W_{12p})$ for all $s\le \frac18\sigma(W_{12p})$, and optimizing over primes gives the $O(s^{4/5}d(s))$ upper bound. If the symplectic Bogomolov-Miyaoka-Yau inequality holds, the same invariants force $\Lambda_s \ge 8s - 1$, so $\Lambda_s/s \to 8$.
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For fixed $C>0$, let $L_C(x)=C x^{1/2}(\log x)^2$ and let $\mu_C(x)=\sum_{x\le n\le x+L_C(x),\ n\text{ composite}} p(n)/n$. The paper establishes that the mean of $\mu_C(x)$ over $x\le X$ equals $4C+O_C(1/\log X)$, its second moment about $4C$ is $O_C((\log X)^{-2})$, and consequently $\mu_C(x)=4C+o(1)$ for almost all $x$. It also proves that the full cumulative sum $S(x)=\sum_{n<x,\ n\text{ composite}} p(n)/n$ has main term $8x^{1/2}/(\log x)^2$, fixing the constant $c=8$ in the old asymptotic. Finally, assuming Hypothesis 1.6—that primes in every interval of length at least $C_0(\log y)^2$ are distributed as $h/\log y$ up to an error $C_A h/(\log y)^A$—the paper proves the uniform estimate $\mu_C(x)=4C+O_C(1/\log x)$ for all $x\ge X_0(C)$, giving an affirmative answer to the Erdős–Graham question. The unconditional proof is elementary and semiprime-based: integers with at least three prime factors contribute only $O(x^{1/3}/\log x)$, so the entire main term comes from semiprimes $pq$ with $p,q\approx x^{1/2}$.
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Theorem 1: if the non-abelian large sieve inequality holds, namely the bound Σ_{q≤Q} Σ_{c,d∈F_q^*} |Σ'_{p≤x} e((p d - i_p c)/q)|^2 ≪ (Q^3+x)π(x), then there exist infinitely many primes p such that 2 is a primitive root modulo p, and #{p≤x: 2 is a primitive root modulo p} ≫ x / log^2 x.
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On the paper's own terms, the discovery is a finite reduction of the two-prime-sum problem to a single numerical condition. For $x=12n+4$, the degree in $G(x/2)$ is $n-f(n)+1$ or $n-f(n)$ according as $x-3$ is prime, where $f(n)$ is the size of the union of the divisibility events $A_p$ for primes $3<p<\sqrt{x}$; analogous formulas are proved for $x=12n+k$, $k=0,2,4,6,8,10$. The paper proves $\prod_{2<p\le y}(1-2/p)\sim e^{-0.183407}/(\log y)^2$, which converts the independent-events approximation into the closed form $\eta(x)$ and shows it is $1.26095$ times half the classical conjectured count. Then, using the definition $\lambda(A,B)=(P(B)-P(A\cap B))/(P(B)-P(A)P(B))$ and $\lambda_0=\max_i\lambda(\bigcup_{j<i}A_j,A_i)$, it proves that whenever $\lambda_0<2.5$ the union probability satisfies $T(n)<1-2/n$ for $x=12n+4$, $n>447$, so the vertex $x$ has more than two neighbours and is a sum of two odd primes; Theorem 6.10 states the same for every sufficiently large even $x$.
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The paper's central claim is that every integer $N\ge2$ can be written as $N=a+b$ with $a,b$ positive integers and $\Omega(ab)\le33$. The proof works with the auxiliary sequence $a_n=n(N-n)$, halved when $N$ is odd, and keeps only indices with $3\nmid a_n$; the prime 3 is then excluded from the sifting set. This pre-sieving removes the extremal $w=3$ case from the dimension condition, allowing a smaller sieve constant $K$. The remainder terms caused by pre-sieving are handled by the inclusion--exclusion identity $r_d=\rho_d-\rho_{3d}+g(d)\rho_3$. The argument then splits into three ranges: the initial range $2\le N\le N_0$ is covered by explicit maximal-prime-gap data, the intermediate range by finite verification of a uniform lower bound, and the large range $N\ge N_1=10^{132}$ by analytic estimates for the main-term factor and the accumulated error term.
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On its own terms, the paper establishes that the restarting Josephus survivor map has no hidden period: $\operatorname{Per}(F_n)=L_n\mathbb{Z}$, proved by freezing trajectories and by using the least recurrence level not dividing a proposed shift as a witness. It then proves that $F_n$ is surjective onto $\{1,\ldots,n\}$: a prime-power CRT construction with blocking primes in $(n/2,n]$ produces intervals of survivor labels, a prime-interval covering criterion joins these intervals, explicit prime estimates handle the tail, and a finite exact certificate closes the remaining range. Finally, it proves a limit law: under a Haar-distributed profinite integer $Q\in\hat{\mathbb{Z}}$, the normalized survivor $X_n=(S_n-1)/(n-1)$ converges to a random variable $Y$ almost surely and in $L^r$ for every finite $r$, the law $\mu$ is symmetric about $1/2$ and nondegenerate, and $\mu([0,\varepsilon])$ and $\mu([1-\varepsilon,1])$ are each at least of order $1/\log(1/\varepsilon)$, which rules out every symmetric $\beta$ law. The paper also proves strict dominance of the endpoint fibers over the two nearest internal positions for every $n\ge 4$, excludes prime levels as minimal counterexamples to endpoint dominance, and verifies endpoint dominance exactly through $n=49$.
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The central claim is Theorem 1.5: assuming the eventual even difference hypothesis (ED), the set of primes cannot be a spectrum of any Borel probability measure. The proof reduces spectrality to two conditions on $a(k)=|\hat{\mu}(k)|^2$: $a(p-q)=0$ for all distinct primes $p,q$, and $\sum_{p} a(n-p)=1$ for every integer $n$. Under ED, the first condition kills all sufficiently large even arguments, which forces $a(k)=1$ for all sufficiently large odd $k$ through the tiling identity at negative odd $n$; then the identity at $n=0$ demands infinitely many terms equal to $1$, an impossibility. A direct corollary, independent of ED, is that any probability measure with prime spectrum would force infinitely many even integers to be absent from the prime difference set.
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The central claim is that short arithmetic progressions cannot cover $1,\dots,n$ when the logarithm of their largest term is much smaller than $\sqrt n$. Formally, any $n$-divisor progression with $\log H = o(\sqrt n)$ has $L \ge (\sqrt{8/27} - o(1)) n^{3/4}/\sqrt{\log n}$. A direct corollary excludes every exponent pair $\alpha<1/2$, $\beta<3/8$, including the originally proposed $(1/3,1/3)$, even under $n^{o(1)}$ slack. The obstruction is sharp at the level of the argument: the constant $\sqrt{8/27}$ arises from optimizing the band $a=2/3$, $b=1$ below $\sqrt n$ and is not claimed to be optimal.
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The central claim is Theorem 1.1: for every fixed $R\ge 1$ there are explicitly computable functions $E_1,\ldots,E_R$ in $C^\infty([0,1])$ such that, uniformly for integers $n\ge m$ as $m\to\infty$, $$\log s(n,m) = m\log n - \psi(m) + \log\frac{m}{n} + m\sum_{r=1}^{R}\frac{E_r(m/n)}{(\log m)^r} + o\!\left(\frac{m}{(\log m)^R}\right),$$ where $\psi(m)=\log \operatorname{lcm}(1,\ldots,m)$. The equivalent form with $m\log n$ replaced by $m\log n - m$ follows from the prime number theorem. The proof factors the residue congruence modulo $L_m$ over primes: primes $p>\sqrt{m}$ act on disjoint coordinate blocks, each block's normalized bias converges to an explicit limit $F_M$, and the prime number theorem converts the accumulated biases into series $A_r(z)$; a moving exponential saddle at $z\approx m/n$ turns these into $E_r$. In the sparse range $n\gg m^2$, an exact binomial identity gives multiplicative asymptotics with coefficients $\kappa_r$, where $\kappa_1=1/4$ and $\kappa_2=(1-\log 2)/4$. The paper also records diagonal values $E_1(1)\approx 0.225872292\ldots$, $E_2(1)\approx 0.046894482\ldots$, $E_3(1)\approx 0.017888502\ldots$, and the boundary identity $E_r'(0)=\kappa_r$.
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The central claim is a two-term asymptotic formula for the Havel–Hakimi residue of the graph $G_n$ on $\{2,\ldots,n\}$ with adjacency $\gcd(x,y)>1$. The paper proves that $$R(G_n)=\bigl(\zeta(2)-1\bigr)\frac{n}{\log n}+\bigl(\zeta(2)-1-A\bigr)\frac{n}{\$log^{2}$ n}+O\!\left(\frac{n}{\$log^{3}$ n}\right),$$ with $A=\sum_{k=2}^{\infty}\log k/(k^2(k-1))=0.3201986326\ldots$. Consequently $R(G_n)\log n/n\to\zeta(2)-1\approx 0.6449$, confirming the constant that had been proposed on numerical and heuristic grounds. The lower bound sums $1/(d(p)+1)$ over prime vertices and applies the theorem that the residue dominates the Caro–Wei sum. The upper bound builds a second graph with the same degree sequence in which almost all low-degree primes are compressed into cliques of the size forced by their degree, with the removed edges restored by switches inside composite classes; since the residue depends only on the degree sequence, the independence number of that auxiliary realization bounds the residue from above.
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Conjecture (S) — every even n ≥ 6 can be written as p + t with p prime and t a twin member — sits strictly between the Goldbach and Dubner conjectures. The paper's central discovery is twofold. First, (S) is a single elementary statement that implies both the Goldbach conjecture and the twin prime conjecture (Theorem 1). Second, any proof of (S) would have to contend with a mod-3 orientation rigidity: for even n not divisible by 3, every twin member appearing in a Goldbach partition of n is a lower member (t+2 prime) when n ≡ 1 mod 3, and an upper member (t−2 prime) when n ≡ 2 mod 3, with the only exceptions being the primes 3 and 5. The paper verifies (S) exhaustively to 10^12, computing th
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The central claim is that the three Mertens sums satisfy explicit inequalities with the same shape as the best current Chebyshev-function bounds. In particular, Theorem 2 proves $|\lambda(x)-\log\log x-M|\le A_\lambda(x_0)(\log x)^{1/2}\exp(-C\sqrt{\log x})$ for every $x\ge x_0$, with $A_\lambda(2)=9.2203$ and $C=0.84768$, and also proves $|\lambda(x)-\log\log x-M|\le A_\ell(x_0)/(\log x)^\ell$ for $\ell=1,\dots,5$. Theorems 3 and 4 give matching bounds for $\Upsilon(x)=\sum_{p\le x}(\log p)/p$ and $\tilde\psi(x)=\sum_{n\le x}\Lambda(n)/n$, with $A_\Upsilon(2)=A_{\tilde\psi}(2)=9.2203$. From these bounds the paper derives two-sided inequalities for the Mertens products $\prod_{p\le x}(1-1/p)
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The paper's central discovery is a rigidity statement at the level of the measure \(\Lambda_{f,t} = \sum_p \frac{f(p)}{p}\delta_{\log\log p - t}\): if \(A_1(X_t;f)\to\kappa\) for \(X_t=\exp(e^t)\), then \(\Lambda_{f,t}\to\kappa\, du\) vaguely as \(t\to\infty\). This is shown by writing \(A_1\) and \(A_2\) as convolutions with the Dickman kernels \(K_1(u)=\rho(e^u-1)\) and \(K_2(u)=\$int_1^{{e^u-1}}$\rho(e^u-1-v)\frac{dv}{v}\), up to \(o(1)\). The convolution identity for \(A_1\) and the nonvanishing of \(\widehat{K_1}\) on the real line imply, by Wiener's Tauberian theorem, that every subsequential limit of \(\Lambda_{f,t}\) is the constant \(\kappa\) times Lebesgue measure. The convolution identity for \(A_2\), together with \(\int K_2 = 1\), then gives \(A_2(X_t;f)\to\kappa\).
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The core claim is that the least quadratic residue modulo n has exponential order 4^k in the number k of odd prime factors, and that this phenomenon is stable enough to transfer to binary quadratic forms. Theorem 1.1 supplies the two sides: a pigeonhole argument shows ℓ(n) ≤ C k² 4^k, while a finite-field construction, translating primes into irreducible polynomials over F₂[x], produces moduli n for which any r with ℓ(n)=r must carry more than 2^k distinct algebraic roots, forcing log r ≥ k log 4 − O(k/log k). Theorem 1.2 sharpens this into a discriminant statement: for any Δ one can build an odd square-free n with log n ≤ C log Δ (log log Δ)² (and under GRH log n ≤ C log Δ log log Δ) such t
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On the paper's own terms, the central discovery is a quantitative lower bound: for sufficiently large n, h_P(n) ≥ n exp((1/50) log n / log log n), and therefore the same inequality holds for F(n). The proof constructs, for a carefully chosen product W of small odd primes, a digit-product set A_t that contains a k-term arithmetic progression of every difference d < W^t (a (k,W^t) arithmetic-Kakeya set) while having size at most W^{t(1−ε)}. Embedding the primes in a CRT-chosen interval then makes the neighborhood of the prime set too small for a matching, so the interval must be long. A separate argument, based on a projection estimate for finite lattice sets, yields F(n) ≤ n^{1.4031+o(1)}; an
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The paper proves that W(u) equals 2|Φ_2(u)|cos(arg Φ_2(u)) plus an explicit error θ_2(u), where Φ_2(u)=Φ(u)(1+α(u)), Φ(u)=exp{−uζ+J(−ζ)} divided by the square root of 2πu(1−1/ζ), J(−ζ) is the incomplete gamma function Γ(0,−ζ), and ζ(u) is the unique solution of e^ζ=−uζ in a specified region, equivalently the 1-branch of the Lambert W function applied to 1/u. The error is bounded by 420 u^{−6} for 6≤u≤19 and by 0.005 u^{−2} for u≥16; a simpler corollary gives |W(u)|<2|Φ(u)| for u≥3. The same method yields a refinement with error below 0.01 u^{−3} on 18≤u≤1000 and below 8.4 u^{−3} for larger u. This is achieved by the saddle point method: the Laplace transform of ω is e^{J(s)}−1, the contour i
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The paper's central claim is the exact identity $\operatorname{str}(\det_p)=p$ for every prime p, together with the asymptotic bounds $\operatorname{str}(\det_n)\ge (1-o(1))n^{0.475}$ and $\operatorname{prk}(\det_n)=n-o(n)$. For prime p, the complement of the determinantal hypersurface in the projective space of p×p matrices is $\mathrm{PGL}_p$, and the argument uses the Chow ring computation $\mathrm{CH}^*(\mathrm{PGL}_p)\cong \mathbb{Z}[x]/(px,x^p)$. If $\det_p$ had a strength decomposition with at most p−1 summands, the summands would define a nowhere-vanishing section of a split vector bundle of rank p−1 whose top Chern class is $(\prod_{i=1}^{p-1} d_i)x^{p-1}$ with each $1\le d_i\le p-1$; this class is nonzero in that ring, whereas a bundle carrying a nowhere-vanishing section must have zero top Chern class. The extension to all n uses a weak monotonicity lemma proved with higher-order directional derivatives, together with the prime-gap bound stating that every large interval of length $x^{0.525}$ around x contains a prime; for partition rank, a stronger monotonicity under column specialization gives $\operatorname{prk}(\det_n)\ge p(n)$, the largest prime not exceeding n, and hence $n-n^{0.525}\le \operatorname{prk}(\det_n)\le n$.
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The central claim is Theorem 1.1: for every nonzero f(x) ∈ Z[x], the set {n > 0 : p_n ∤ f(n)} is infinite. The paper shows this by contradiction: assuming p_n | f(n) for all large n, it uses the bounded-gaps theorem to find infinitely many n for which a fixed block of k+1 consecutive primes has fixed offsets h_0,...,h_k. It then constructs a nonzero polynomial Φ(x,y) with integer coefficients such that, on that infinite set, Φ(n,p_n)/ρ(p_n) is an integer that is also o(1), forcing Φ(n,p_n)=0 eventually, which contradicts the leading-term growth of Φ. Since the existence of such f would mean π(p) satisfies a nontrivial polynomial relation over Q, the conclusion is that π(p) ∉ C_A, the integra
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Theorem 1.4 asserts that as x→∞, the count of n<x with P^+(n)<P^+(n+1) exceeds 0.280x, and by symmetry the same holds for the reverse inequality. The constant comes from a decomposition of n according to the sizes of P^+(n) and P^+(n+1): a cutoff c=0.1348 separates small and large cases, and each piece is estimated either by the standard smooth-number distribution or by a new lower-bound function C(η) for smooth numbers in arithmetic progressions. With the choices δ1=0.417 and an optimized auxiliary function t1(η), the assembled lower bound C(c,δ1)−2ν is reported to exceed 0.280.
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The paper's central claim is that the Skolem Problem reduces to a question about prime gaps. It defines good primes as those that are not 'bad', where badness means that a prime divides the algebraic norm of a certain exponential-polynomial expression associated to a small LRS around a potential large zero. For any large zero of an LRS, the paper shows there must be an interval around the zero containing far fewer good primes than the prime-gap conjecture would predict; if the conjecture holds for good primes, this contradiction rules out large zeros. The unconditional theorem shows that across all LRS, the set of large-zero indices is so sparse that its complement still allows an algorithm
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For every digit d from 0 to 9, the total number of occurrences of d among all digits of all primes less than 10^n, divided by the total number of such digits, equals 1/10 plus an error of size O((log n)/n) as n tends to infinity, uniformly in d. The same limit therefore holds in total variation for the empirical digit measure.
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For beta-sieve weights (or suitable Lambda-squared weights) the sum U = sum_d phi(d) (sum_{m: d|m} lambda_m/m)^2 is << product_p (1-1/p). This is the same order of magnitude previously obtained for the larger sum W, and is precisely the quantity needed for the short-interval applications.
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For each r > 2 there exist constants r' < c(r) < 2 < C(r) with lim c(r) = 1 and lim C(r) = ∞ such that the r-variation of the prime-weighted Carleson series is bounded on ℓ^p for all c(r) < p < C(r), while the variation is unbounded for p ≤ r'. The same reductions give that the maximal prime Carleson operator is bounded on ℓ^p(ℤ) for every 1 < p < ∞.
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Guth and Maynard prove a new large-values estimate for Dirichlet polynomials that yields the zero-density bound N(σ,T) ≪ T^{15(1-σ)/(3+5σ)+ε} uniformly for 1/2 ≤ σ ≤ 1; this is the first improvement on Ingham's 1940 estimate throughout the range σ ≤ 3/4 and, when combined with earlier results, gives A(σ) < 30/13, which in turn implies the prime-number theorem in short intervals of length x^{17/30} and almost all intervals of length x^{2/15}.
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For every fixed k≥2 and every N≥0 the additive Mertens sum admits the complete expansion Sk(x)=(x^{k-1}/log^k x)∑_{n=0}^N Ek,n/log^n x + O(x^{k-1}/log^{k+N+1}x), where the coefficients are the absolutely convergent integrals Ek,n=(-1)^n ∫_{(0,1]^k} hn(log t1,…,log tk)/(t1+…+tk) dt and hn is the complete homogeneous symmetric polynomial of degree n.
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Assuming the Riemann Hypothesis and a mild lower bound on the Gaussian smoothing width relative to a fixed simple critical-line zero, the full Gaussian–Perron prime-force defect near that zero equals the universal selected-zero profile −a Re(e^{−λ}/λ) plus an error that is O(1/log X) plus exponentially small, uniformly on compact sets of the logarithmic displacement λ away from zero.
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On the paper's own terms, the central discovery is that Gilbreath's conjecture is equivalent in effect to a balance property of the right diagonal of the absolute-difference triangle. Once the triangle's rightmost column reaches a long enough stretch of 0s and 2s, any subsequent term that is not too large keeps the bottom cell equal to 1. Theorem 5.5 makes this quantitative: with g*_n < n^α and ν2(q_{n−1}) > n^β and β>α, a valid successful sequence cannot fail at the next step. Applying the known prime-gap exponent α=0.525 (Baker 2001) and conjecturing β=0.99 for the primes, the author obtains what he describes as a trivial proof of Gilbreath's conjecture if the conjecture holds, and offers
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For a Serre curve — an elliptic curve over Q whose adelic Galois image is as large as possible, namely the index-2 subgroup forced by the quadratic field Q(√Δ_E) — Theorem 3 provides a closed formula for C^{m-div}_E, the natural density of good primes p with m | #E_p(F_p). Writing m = m_1 m_2 with m_1 = gcd(m, m_E^∞), the density equals the average density C^{m-div} when m_1 is outside a small exceptional 2-adic range; otherwise it is (C^{m1-div} + ∏_{ℓ^α∥m1} −ℓ^{2α−1}/|GL_2(Z/ℓ^αZ)|) · C^{m2-div}. This formula yields Corollary 4: C^{m-div}_E > 1/m for every m ≥ 2. Theorem 5 then shows that the k-th power average of |C^{m-div}_E − C^{m-div}| over the box family tends to zero as the box grows
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The main result, Theorem 2.2, states that if n=pq and n+12=p'q' are both products of two distinct primes with p<q and p'<q', then the two jump conditions φ(n+12)-φ(n)=12 and σ(n+12)-σ(n)=12 hold together if and only if p≥5, q=p+8, and {p',q'}={p+2,p+6}. The proof reduces both equations to the equality p+q=p'+q', then uses the discriminant relation (q-p)^2-(q'-p')^2=48. Factoring 48 gives only three possible pairs of square gaps; parity rules out all but (8,4), forcing the prime-quadruplet spacing. A separate theorem shows no prime power p^k with k≥2 can satisfy the first jump condition alone.
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We exhibit an explicit integer M < 1.78 × 10^{32} such that every integer m ≥ M with gcd(m,9)=1 occurs as s(p) for at least one prime p. The proof uses explicit major-arc estimates, a fully explicit replacement for the implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates.
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We prove unconditionally that Proposition (1+1.9) is true. Thus every sufficiently large even integer N can be written as N = p + r q, where r ≤ q^{0.9}, r is 1 or prime, and p, q are primes. Assuming the Elliott-Halberstam Conjecture the exponent 1.9 can be replaced by 1.4. The same method yields Proposition (1-1.75) for the twin-prime side unconditionally and (1-1.4) under Elliott-Halberstam.
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For every k ≥ 1 the number of primes P_k in the interval I_k between consecutive odd squares equals N_k − S_k + E_k, where N_k = 4k is the number of odd integers in I_k, S_k is the total matrix multiplicity of those integers, and E_k corrects for excess multiplicity of non-semiprime composites. Consequently a prime exists in I_k if and only if the combinatorial inequality E_k ≤ S_k − N_k holds.
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On the paper's own terms, the central discovery is Theorem 2.5: g_n < (51/16) log² p_n for every n ≥ 1. The engine is a set of summation identities (Theorem 2.1) expressing each gap g_n through B_n = sum_{k=2}^n g_k/(k−1) and T_n = sum_{k=2}^n A_k/(k−1) + (p_n−2)/n − A_1, where A_k is the average of the first k gaps. Lemma 2.2 asserts that B_n < 2B_{n−1} for every n ≥ 22; Lemma 2.3 converts that into g_n < (51/16) B_{n+1}; and explicit classical bounds on the nth prime, n log n < p_n < n(log n + log log n), let the author bound B_{n+1} by a logarithm-squared expression. The cases n ≤ 21 are checked directly.
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The positivity of Cojocaru's constant C_{E,j} — which controls the density of primes where the smallest invariant factor of E(F_p) is j — can be studied through the image of the adelic Galois representation of E, and experimentally the constant vanishes precisely when division fields of E coincide. This reframes an analytic density question as a group-theoretic and field-arithmetic one, and the authors classify several coincidence families arising from abelian division fields.
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In the paper's own terms, the discovery is that Chebyshev's bounds follow from comparing two weighted prime sums: π(x) ≍ (∑_{p≤x}√(ln p/p))² − ∑_{p≤x} ln p/p, and (∑_{p≤x}√(ln p/p))² ≍ x/ln x. The second statement is shown by combining Abel's identity with Mertens' estimate; the first is shown by writing the difference E(x) at prime indices and using the exact recurrence E_n = E_{n-1} + 2a_n S_{n-1}, where a_n = √(ln p_n/p_n). Since S_n ≍ 1/a_n, each increment 2a_n S_{n-1} is bounded above and below by positive constants, forcing E_n ≍ n = π(p_n). Thus the prime count is sandwiched twice, yielding the Chebyshev bounds.
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The paper establishes an asymptotic formula for the average of the error term appearing in Bateman–Horn’s conjecture, taken over an exponential range of arguments; the resulting main term is strictly larger than square-root size, so the averaged error exhibits worse-than-square-root cancellation.
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On the paper's own terms: Mersenne prime exponents have elevated divisor complexity of p-1 relative to local prime controls, and this elevation survives matching by exponent size. The mean percentile rank of S(p) in a 5000-prime window is 0.648 versus a null expectation of 0.5; sign, Wilcoxon, and permutation tests reject the null with p-values around 0.002. The paper interprets this as evidence for a structural refinement of the standard probability estimate for M_p being prime, preserving the log p / p scale after averaging over S.
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Informational cardinality I(M)=(α(M),δ(M),ι(M)) distinguishes the essential fractal prime set P_ess from the generalized Cantor set C_{1/3}: both have continuum cardinality (α=1), yet dim_H(P_ess)=1/2>1/3=dim_H(C_{1/3}) and ι(P_ess)=-ζ(1/2) while ι(C_{1/3})=0, so I(P_ess)>I(C_{1/3}) under lexicographic order. The same framework pairs P_ess with a fractal zero set Z_F of equal dimension and conjectures that their information measures sum to zero.
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The central discovery is that a deterministic fractal set, PF, can be built directly from the prime residues modulo 16 and that its fractal dimension is exactly 1/4, with Hausdorff and box-counting dimensions agreeing. At each stage, every interval is split into 16 equal pieces and only two are kept, those indexed by the current prime residue a_n and its opposite a_n+8 mod 16. The proof shows that the number of length-16^{-n} intervals needed to cover PF is exactly 2^n, giving box dimension log 2/log 16. For the Hausdorff dimension, the paper constructs a natural mass distribution and proves a measure-decay estimate, then uses the mass distribution principle to get the lower bound; the upper
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On the author's own terms, the paper establishes that primes have an average distribution in arithmetic progressions compatible with moduli beyond sqrt(x). In particular, Theorem 1.1 asserts that for composite moduli q = q1 q2 with Q1^2 Q2 < x^{1-epsilon} and Q1^7 Q2^12 < x^{4-epsilon}, any divisor-bounded bilinear weight gives a discrepancy sum of size O(x/(log x)^A). Since those two constraints are compatible up to q1 q2 = x^{9/17}, this is a bilinear mean-value theorem at modulus size x^{9/17}; the analogous trilinear statement reaches x^{17/32}. The paper also constructs explicit functions rho0(n) <= 1_p(n) <= rho1(n) whose sums and arithmetic-progression distributions are controlled, an
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Theorems 1.1 and 1.2 assert that for h = X^θ with 2/15+ε < θ < 0.99, all but O_ε(X exp(-c_ε (log X)^{1/4})) integers x ∈ [X,2X] satisfy R(x) ≍ h and S(x) ≍ h, where R(x) counts representations of n ∈ (x,x+h] as p+a (p prime, a in a lacunary set A_λ(X) of size ≍ log X) and S(x) counts distinct such n. The lower bound R(x) ≫ h comes from a quoted almost-all primes-in-short-intervals asymptotic (restricting to a ≤ h/2); the upper bound on the second moment Q(x) ≪ h comes from an upper-bound sieve for prime pairs plus a structural estimate on small prime factors in differences a_1-a_2; Cauchy–Schwarz then gives S(x) ≥ R(x)²/Q(x) ≫ h.
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On the paper's own terms: a proposition P is α-decidable if there exists a Brouwer ordinal y such that P ↔ y ≥ α. Using the type Brw of Brouwer trees, whose limit constructor accepts only strictly increasing sequences, the paper shows that 1-decidability coincides with ordinary decidability and (ω+1)-decidability with semidecidability. The main theorem (Theorem 7.5) states that for a family P : N → Prop of semidecidable propositions, the countable meet ∀n.Pn is ω²-decidable. The proof constructs a characteristic ordinal Ψ(P) for such a family and establishes the equivalence ∀i.Pi ↔ Ψ(P) ≥ ω², relying on Lemma 7.7, which says that dropping a finite prefix does not push the characteristic ordi
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The central claim is Theorem 1.4: along the sequence of primes p_n, the return-times averages converge pointwise for all second systems, and the Wiener–Wintner averages converge for each fixed frequency θ. The mechanism is to write the prime indicator through the von Mangoldt function, approximate it by a truncated Heath–Brown model, control the U^3 norm of the part of the model with fixed rational complexity Q by the bound Q^{-3/8+o(1)}, and then transfer Gowers-norm control of the weight to almost-everywhere convergence of weighted ergodic averages via uniform Wiener–Wintner and weak-mixing reductions.
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The load-bearing result is Theorem 1: for all 2≤n≤4,553,432,387, every row of the Sierpiński matrix S_n contains at least one prime. The proof works because the largest prime gap starting below (4,553,432,387)² is the 82nd maximal gap, of width 1,572, so the pigeonhole principle guarantees at least floor(n/1,572) primes in each row for n in that range. The small cases n≤13 are checked directly, or by a known 1961 verification used to cover the overlap n≤4,505; the two ranges splice together to cover all n up to N₈₃. For n beyond N₈₃, Theorems 3 and 7 provide guarantees that at least n/4 rows, and the bottom 131,294 rows, contain primes.
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If α is an irrational number having a continued fraction expansion with bounded terms, then the number of primes p in the interval (X-Y,X] satisfying ||pα||<δ is asymptotically equal to 2δ Y/log X, provided that X≥10, X^{2/3+ε}≤Y≤X/2 and X^ε max{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}}≤δ≤1/2.
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The central claim is that a scaled omega-Gibbs prior on prime multipliers, pi_p proportional to 2^{-beta * ell_omega(p)} with beta > 1, makes the MTE tractable and universal: additive gaps X_{t+1} - X_t have asymptotically Pareto tails with exponent beta, and the time-averaged omega codelength converges almost surely. The pure beta = 1 prior is a critical boundary case: it is regularly varying with index 1, has infinite first moment, and lies outside the theorems. On Debian and PyPI package-size data, the fitted slope a is below log 2, implying beta < 1, which the authors read as the signature of human-generated, non-averaging complexity.
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Barrett's formula asserts that Barr(n) = 3 + sum_{k=5}^{n-1} floor( sin(pi (k-1)!/k) / sin(pi/k) ) equals the number of primes p < n, under the historical convention that 1 is counted as prime. The summand is a geometric indicator of primality: for composite k ≥ 5, (k-1)! is divisible by k, so the numerator's sine is zero; for prime k, Wilson's theorem forces the ratio to be 1. Thus every integer k contributes exactly 1 if k is prime and 0 otherwise, making the formula a finite, exact trigonometric expression for the prime-counting function.
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The paper's core discovery is Theorem 8.2: RH holds if and only if the trajectory error functional satisfies E(X) ≪ X^(1/2) log X for all X ≥ e^120. The proof runs through a chain of unconditional estimates—one-visit and parent-window lemmas limiting how often a trajectory hits a window, macro-step alignment showing that L ≍ log X composite steps contract the scale from X to X^(3/4), and a frequency-netting lemma that controls the sum over Riemann-zero contributions using an explicit smoothed formula with cubic-log truncation and a grid-based large sieve. Iterating the contraction inequalities yields the unconditional bound E(X) ≪ X^(1/2) log X (Corollary 6.6), and the paper then argues that
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Theorem 2.6 of the paper is that the sliding-window 'puzzle' algorithm computes h(k,x) on input x in O(x log x) arithmetic operations and O(x^(2/3)) space. The priority-queue algorithm, which keeps one chain of each length in a min-heap keyed by sum, achieves O(x log x) time and O(x^(3/5)) space when primes are produced by an incremental interval sieve, or O(x(log x)^2) time and O(sqrt(x log x)) space when primes are produced by a pseudosquares prime test. The analysis is average-case over prime gaps: the basic estimate, Lemma 2.3, is that sliding a chain of length m with sum n changes the sum by Theta(m log(n/m)) on average. The implementations ran to x = 10^14 + 145300 (the paper notes tha
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The paper's central claim is that old existence theorems for primes in short intervals can be made fully explicit by splicing together three independent effective inputs. Using a cited 90th-power theorem, interpreted by inspection as [x, x + 90 x^(1 - 1/90)] for all real x >= 1, plus the cited cube theorem with its improved threshold, plus the known table of maximal prime gaps and explicit linear interval data, the paper derives three effective statements: for n >= 4 and x >= exp(exp(33)) the interval [x, x + x^(1 - 1/n)] contains a prime; for n >= 91 and x >= [90^90]^(n/(n-90)) it contains a prime; and for n >= 106 the statement holds for every x >= 1. The last statement yields the universa
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The central claim is that pointwise ergodic averages along the set of polynomial values of primes of the form x^2 + n y^2 converge almost everywhere for every function in L^p with p > 1, for every measure-preserving system. The proof adapts the Hardy-Littlewood circle method to this arithmetic setting, with the novel ingredient being estimates for the distribution of the prime ideals that correspond to the quadratic form. The paper further constructs an L^1 function for which these averages fail to converge, proving that the p > 1 condition is optimal.
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The central claim is that for an irrational $\alpha$ and fixed $\tau\in(0,1/8)$, the set $\mathcal{P}_\tau(\alpha)=\{p\text{ prime}:\lVert\alpha p\rVert\le p^{-\tau}\}$ contains infinitely many nontrivial three-term arithmetic progressions. The novelty is that membership depends on a single irrational multiplier and the selected primes become sparser as $p$ grows, yet additive configurations of length three still occur infinitely often. The paper further states a binary Goldbach-type result for sums of two primes from the same thin Bohr set.
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The core discovery is a finite counterexample: the analogue of the second Hardy-Littlewood conjecture for cyclic numbers is false. In its prime form, that conjecture says the number of primes in an interval $(x,x+y]$ never exceeds the number of primes up to $y$. Replacing primes by cyclic numbers gives the inequality $C(x+y)-C(x)\le C(y)$, where $C$ counts cyclic numbers; the paper reports that this inequality fails somewhere among cyclic numbers below $10^8$. The enumeration total is $28{,}488{,}167$, and the equivalence $n$ cyclic $\iff \gcd(n,\phi(n))=1$ is what makes the search purely arithmetic.
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The prediction is that in almost all prime gaps $(p_n,p_{n+1})$, there is a natural number $m$ with least prime factor $p(m) \ge p_{n+1}-p_n$. The paper establishes this prediction. Let $N(X)$ count exceptional $n$ with $p_n \in [X,2X]$, meaning no such $m$ lies in the gap. The main unconditional result is $N(X) = O(X/\log^2 X)$. Conditionally on a form of the prime tuples conjecture, the paper obtains $N(X) \sim cX/\log^2 X$ for an explicit constant $c>0$ numerically suggested to be about $2.7$--$2.8$. The proof is a sieve-theoretic argument; the full-strength version depends on a previously developed asymptotic for singular series.
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The central claim is that $f_{Z,k,H}(n)=n^2-(2Zk-1)n+\frac{(2Zk-1)^2+H}{4}$ is a prime-generating quadratic whose behavior can be controlled by the pair $(Z,k)$ while $H$ ranges over the Heegner numbers. The paper states that through specific substitutions this form is closely related to the Euler-Rabinowitsch polynomials, so it both recovers known prime-rich quadratics and adds extra tuning freedom. Using the Bateman-Horn conjecture and prime-counting functions, the paper asserts that the family can be optimized to generate a high density of primes, and takes this as evidence that Heegner numbers have a concrete impact on the distribution of primes in quadratic sequences.
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The central claim is that $N(\sigma,T)$ is $O(1)$ uniformly in $T$ when $\sigma$ is sufficiently close to $1-\nu(t)$, the left edge of the Korobov-Vinogradov zero-free region, where $\nu(t)=A_0(\log t)^{-2/3}(\log\log t)^{-1/3}$. Exploiting this uniform boundedness in the minimization that defines $\omega(x)$, the paper obtains $\psi(x)-x \ll x\exp\{-\omega(x)\}$ with no $\varepsilon$ slack, i.e. $\varepsilon=0$ in the stated bound.
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The central claim is that two perturbations of Maynard's sieve raise the optimal ratio $M(F)=I(F)/J(F)$ by an additive amount $\delta/2 + \epsilon\ln\ln k$ beyond the baseline $\frac{1}{4}\ln k$. The first perturbation enlarges the simplex $R=\{t\in[0,\tau]^k:\sum t_i\le 1\}$ to $R'=\{t\in[0,\tau]^k:\sum t_i\le 1+\delta\chi(\sum t_i)\}$, where $\chi$ is a logistic-map iterate, and the second replaces $F$ by $F+\epsilon\xi$ with $\xi(t)=\prod_{j=1}^k\Phi(t_j)$ built from the normal CDF as a stand-in for GUE spacing statistics. Theorem 4.3 encodes the resulting heuristic formula $M'(F')\sim\frac{1}{4}\ln k+\frac{\delta}{2}+\epsilon\ln\ln k$. The paper then uses the standard sieve criterion $M>m \Rightarrow$ infinitely many $m+1$ primes in an admissible $k$-tuple to claim $M'>3$ at $k\approx 40$, yielding 4-prime blocks and hence gaps of at most 180 unconditionally and at most 8 under a partial Elliott–Halberstam assumption with $\theta=0.8$.
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Theorem 1.10: For any function φ : N → R_{>0} and ℓ ∈ N, we have dim_H E'_ℓ(φ) = dim_H E_ℓ(φ). If correct, this means restricting infinitely many partial quotients to be prime does not change the Hausdorff dimension of the set where products of ℓ consecutive partial quotients exceed φ(n). The paper also claims the zero-one law in Theorem 1.8: m(E'_ℓ(φ)) = 0 or 1 according to the convergence or divergence of ∑ (log log φ(n))^{ℓ-1}/(φ(n) log φ(n)).
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The central discovery is that the log-primorial sequence $\vartheta_n = \ln([p_n]\#)$ has provably regular gaps: $\vartheta_{n+1}-\vartheta_n = \ln p_{n+1}$, so the relative gap lies in the narrow band $9/(10n) < \mathfrak{g}_n/\vartheta_n < 9/(5n)$, and also $(22/25)\ln p_n/p_n < \mathfrak{g}_n/\vartheta_n < 2\ln p_n/p_n$. From these gap bounds, the paper derives explicit upper and lower bounds on the counting function $\pi_\vartheta(x) = \#\{i : \vartheta_i \le x\}$, then uses them to prove $\vartheta$-analogues: $\vartheta$-Andrica holds with maximal step $\sqrt{\ln 6}-\sqrt{\ln 2}\approx 0.506 < 1$; $\vartheta$-Firoozbakht holds for $n\ge 4$; $\vartheta$-Fourges holds for all $n\ne 2$; $\vartheta$-Nicholson holds for $n\ge 3$; $\vartheta$-Farhadian holds for $n\ge 5$; and $\vartheta$-Legendre, $\vartheta$-Oppermann, and $\vartheta$-Brocard follow from the counting bound $\pi_\vartheta(x)-\pi_\vartheta(y) > (x-y)/(\ln x + 1.059660101)$. The paper stresses that this does not transfer to the ordinary prime conjectures, since the identity $p_n = \exp(\vartheta_n-\vartheta_{n-1})$ does not naively convert $\vartheta$-gap theorems into prime-gap theorems.
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On the paper's own terms, the central claim is that the map from zero-free regions to prime-number-theorem error terms is captured, up to the value of one constant, by the variational quantity $\omega(x)$. Theorem 3.1 shows that for a Beurling zeta function satisfying Axiom A (generalized integers counted by $Ax+O(x^\theta)$) and a zero-density estimate $N(\sigma,T)\ll T^{A(1-\sigma)^B}(\log T)^C$, any zero-free region $\sigma>1-f(\log|t|)$ inside the regular/slow variation class forces the explicit upper bound displayed in the summary; with the currently best Beurling zero-density estimate this becomes the concrete bound of Theorem 1.5. Theorem 1.6 shows the converse construction: for every admissible $f$ with $1/u=o(f(u))$, there is a Beurling number system with $N_P(x)=Ax+O_\varepsilon(x^{1/2+\varepsilon})$, with infinitely many zeros on $\sigma=1-f(\log|t|)$ and none to the right, and with $\psi_P(x)-x=\Omega_\pm(x\exp(-\omega(x)+(1-\delta)f(u_0(x))u_0(x)))$ for every $\delta>0$. The paper also proves a destructive-interference variant showing the parameter $\varepsilon$ in the oscillation theorem cannot be set to zero, and identifies that the classical method for deriving zero-free regions from growth is sharp up to a constant among extended Beurling systems.
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The paper shows that for certain pairs of normalized error terms, such as those associated with ψ(x) and the weighted sum involving Λ(n)/n, the inequality between them holds for all sufficiently large x if and only if the Riemann hypothesis is true. Assuming both the Riemann hypothesis and the linear independence of the positive imaginary parts of the zeta zeros, the authors calculate that the logarithmic density of the set where two specific error terms have the same sign is approximately 0.9865.
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On the paper's own terms, the central claim is Theorem 3: for $g \ge 2$, $L \in \mathbb{N}$ and $2 \le x \le g^L$, the exponential sum $S = \sum_{n \le x} \Lambda(n) e(f_L(n))$ satisfies $S \ll x g^{-\kappa} (\log x)^4$, where $\kappa = \frac{1}{10} \sigma_\xi(\alpha)$ and $\xi = \lfloor \tfrac{1}{4} \log x / \log g \rfloor$. Theorem 2 then gives the quantitative count of reversed primes of length $L$: writing $\pi_L^-(a,q)$ for the number of $p \in [g^{L-1}, g^L)$ with $\mathrm{rev}(p) \equiv a \pmod q$, one has $\pi_L^-(a,q) = \frac{\rho_g(a,q)}{q} \frac{g^L}{\log g^L}(1 + O(1/L)) + O(g^L \exp(-c \sqrt{L}))$ whenever $q \le \exp(c \sqrt{L})$ and $(a,q,g^2-1)=1$ with $g \nmid (a,q)$. The explicit factor $\rho_g(a,q)$ encodes the necessary coprimality conditions, and the error term is effective, depending only on $g$.
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On the paper's own terms, the central discovery is that a recent large-value method for long Dirichlet polynomials survives the passage to polynomials twisted by primitive Dirichlet characters, provided the character sums are controlled by a new bound for affine transformations with GCD twists. For a separated set $W$ of pairs $(t,\chi)$ on which a character-twisted polynomial of length $N$ attains values $\ge V$, Theorem 1.3 gives $|W|\ll_\epsilon N^2V^{-2}+(qT)^{4/3}N^2V^{-4}$ when $(qT)^{3/4}\le N\le (qT)^{5/6}$, and a further four-term bound for larger $N$. Fed into the zero-detection method, this yields $\sum_{\chi\bmod q}N(\sigma,T,\chi)\ll_\epsilon (qT)^{4(1-\sigma)/(1+\sigma)}$, and the combination with the classical bound for $\sigma\le 5/7$ produces the headline estimate $(qT)^{7(1-\sigma)/3+\epsilon}$. The paper also shows the new exponent pays off arithmetically: for fixed prime $p$, the least prime $p(p^n,k)$ in the progression $k\bmod p^n$ is $\ll_{p,\epsilon}(p^n)^{7/3+\epsilon}$, and the least Goldbach number $G(p,k)\equiv k\pmod p$ is $\ll_\epsilon p^{7/6+\epsilon}$.
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The central claim, stated as Eq. (5), is that the twin prime counting function can be written as $\pi_2(x)\sim \frac{2x}{(\ln x)^2}\cdot \frac{\sum_{t=0}^{\infty}(-2)^t f(t;z)}{(\sum_{t=0}^{\infty}(-1)^t f(t;z))^2}$, where $f(t;z)$ is the degree-$t$ elementary symmetric polynomial in the reciprocals of odd primes up to $z$. The derivation expands the numerator product $\prod_{3\le p\le z}(1-2/p)$ as $\sum_{t=0}^{\infty}(-2)^t f(t;z)$ and the denominator product as $\sum_{t=0}^{\infty}(-1)^t f(t;z)$, so the ratio reproduces $D(z)$ term by term. This recasts the sieve correction factor, and ultimately the Hardy-Littlewood constant, as a structured alternating series instead of an opaque product. The paper's own numerical test, with $z=\lfloor x^{1/4}\rfloor$ and terms through $t=4$, is not accurate for larger $x$; the author attributes the systematic overestimation to truncation and presents the framework as a constructive heuristic whose convergence merits further analysis.