{"id":"95f7f16c-c436-420a-8706-382a46a285c6","arxiv_id":"2507.13780","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For a broad class of zero-free regions, the paper sharpens the prime number theorem error term and constructs Beurling systems showing this sharpening is nearly optimal.","lead":"This paper proves a sharper version of the known connection between where the Riemann or Beurling zeta function has no zeros and the size of the error in the prime number theorem. It also constructs special generalized prime systems whose zeta zeros sit exactly on a prescribed curve, showing the new error bound is nearly optimal.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"I read the full manuscript in good faith. The central claims are the upper-bound theorem (Theorem 1.5 / Theorem 3.1) and the matching-oscillation construction (Theorem 1.6). The proofs follow established templates from Pintz, Revesz, Johnston, and Diamond-Montgomery-Vorhauer, with the Beurling zero-density estimate and random approximation supplied by cited prior work. I checked the main technical steps: the decomposition of the explicit formula into S1, S2, S3; the use of zero-density estimates to control small and large imaginary parts; the phase-alignment of planted zeros to produce constructive interference; and the discrete approximation via the author's random-approximation theorem. I found no fatal gap. The weakest point is exactly what the reader noted: the class of admissible f (eventually C^1, decreasing, strictly convex, regularly or slowly varying with technical conditions) is restrictive, but the author explicitly acknowledges this and the theorems are stated for that class, so it is not a correctness flaw. The two proof-presentation issues I found (the (3.3)-versus-(3.2) citation in Theorem 3.1 and the (1+alpha) factor slip in Section 5.5) are correctable without changing the results. Since the mathematical arguments hold up under scrutiny, I see no reason to alter the reader's ACCEPT verdict.","tokens_in":28805,"tokens_out":48268,"duration_ms":524460,"concrete_test":"Recompute the asymptotic in Section 5.5 from (2.3) and (2.4), replacing the printed (1+alpha) factor by (1+alpha)^(-1), and verify that the inequality M_k >= exp((1-2delta) u0(x_tilde_k) f(u0(x_tilde_k))) still holds; if it does, the oscillation proof is intact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing mathematical flaw found. Theorems 1.5 and 1.6 are supported by the written arguments; the central zero-counting, contour-planting, and discretization steps are coherent. Two presentational issues do not threaten the claims: (i) in the proof of Theorem 3.1, the S1 and S3 estimates invoke the concrete Beurling zero-density estimate (3.3) instead of the stated general hypothesis (3.2), but the same epsilon-absorption argument works under (3.2) after choosing epsilon small relative to delta; (ii) in Section 5.5, the displayed asymptotic f(u0(xt)) ~ (1+alpha) omega(xt)/log xt is a typo, since (2.3) gives f ~ (1/(1+alpha)) omega/log x. The intended conclusion f(u0(xt)) ~ f(u0(xk)) is correct, and the subsequent lower-bound comparison remains valid. The acknowledged restriction to regularly or slowly varying f delimits the theorem's scope but is not an internal inconsistency.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the relation between zero-free regions of Beurling zeta functions and the error term in the prime number theorem. Assuming Axiom A (1.1) and a zero-density estimate (3.2), Theorem 3.1 gives an upper bound for ψP(x)−x of the form x exp(−ω(x) + (A+δ)f(u0(x))^B u0(x)) ω(x)^C for zero-free regions σ > 1 − f(log t), where f is eventually C^1, decreasing, strictly convex, and either regularly varying of index −α (0<α≤1) or slowly varying with conditions (1.9)–(1.10). This refines the ε-loss in Pintz–Révész to a decaying secondary term. Theorem 1.6 provides a converse construction: for every such f with 1/u = o(f(u)), there exists a Beurling number system whose zeta function has infinitely many zeros on the contour σ = 1 − f(log|t|), none to the right, and whose PNT error oscillates at the complementary level x exp(−ω(x) + (1−δ)f(u0(x))u0(x)). The paper also includes variants: a Lindelöf-type construction with only xe^{−ω(x)} oscillation, an example showing ε cannot be zero in the converse Pintz theorem, and remarks on Landau's method and zero clustering.","tokens_in":28874,"tokens_out":14293,"duration_ms":145637,"significance":"The main results give the sharpest known quantitative connection between zero-free regions and PNT error terms in the Beurling setting, improving on Pintz, Johnston, and Révész by removing the arbitrary ε in the exponent at the cost of a secondary term. The constructive half of the paper is a substantial technical achievement: it verifies positivity of the prime measure, bounds the resulting zeta function, locates all zeros, and derives matching oscillation estimates, building on the Diamond–Montgomery–Vorhauer method and the author's random approximation theorem. The paper also gives useful structural insights, showing that zero-free regions alone do not imply improved zero-density estimates, and that Landau's method is nearly sharp for a class of Beurling zeta functions. The restriction to regularly or slowly varying f is explicitly acknowledged and covers essentially all zero-free regions used in practice.","major_comments":[],"minor_comments":[{"comment":"The estimates for S1 and S3 use the concrete zero-density estimate (3.3) with exponents 5ε/(1−θ), while the theorem is stated under the general hypothesis (3.2). The proof should be modified to use (3.2) directly, choosing ε sufficiently small relative to δ and absorbing the logarithmic factor; this is a presentational issue, but as written the proof does not cover the generality claimed.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The sentence \"Suppose first that f is slowly varying of index −α, 0 < α ≤ 1\" should read \"regularly varying\" because the subsequent use of (2.3) is the regular-variation asymptotics.","section":"Section 3, first paragraph after the definition of S2"},{"comment":"The approximation f(u0(xt)) ∼ (1+α) ω(xt)/log xt is reversed; (2.3) gives f(u0(xt)) ∼ ω(xt)/((1+α) log xt). The intended conclusion f(u0(xt)) ∼ f(u0(xk)) is unaffected.","section":"Section 5.5, displayed relation before the final comparison"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is well within the scope of the journal and the self-citations to [5] and [7] are appropriate since the quoted theorems are independent published results. The only issues are local and can be addressed in a revision; I would be happy to see the revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the full paper. The main new things are Theorem 1.5 and Theorem 1.6. The first improves Johnston's recent result for the natural class of regularly or slowly varying f and extends it from the Riemann zeta function to general Beurling systems, replacing the epsilon loss with a secondary term of size f(u0)^B u0 times omega^C. The second generalizes the Diamond–Montgomery–Vorhauer construction from f(u) = c/u to arbitrary such f, plants zeros exactly on the prescribed contour with none to the right, and gets an oscillation lower bound that matches the upper bound up to the constant. That is a genuine step forward on Ingham's question.\n\nThe upper-bound proof is a careful adaptation of the Pintz–Révész template with dyadic splits; I found the estimates coherent. The construction in Section 5 verifies positivity of psi_c', bounds the zeta function, locates the zeros, and derives the oscillation estimates; the random approximation to pass to a discrete system is standard and correctly applied. I agree with the reader's ACCEPT verdict and with the stress-test note: no load-bearing flaw.\n\nSoft spots, in proportion. The theorems are restricted to f that is eventually C^1, decreasing, strictly convex, and regularly or slowly varying. The author flags this as the most restrictive assumption, and it is what makes u0(x) unique and gives (2.3)–(2.4). Outside that class you only get Révész's epsilonic versions, so the scope is delimited but not internally broken. Two small presentational issues in the written proof: in Theorem 3.1 the S1 and S3 estimates invoke the concrete zero-density estimate (3.3) rather than the general hypothesis (3.2), though the same epsilon-absorption works under (3.2); and in Section 5.5 the displayed asymptotic f(u0(x_t)) ~ (1+alpha) omega/log x is a typo for the reverse ratio. Neither affects the claims.\n\nThe self-citations of [5] and [7] are fine: those are independent published results and do not smuggle in the target error-term bound.\n\nWho this is for: analytic number theorists working on Ingham's problem, Beurling generalized primes, and zero-free regions. It deserves a serious referee; I would send it to a specialist and expect the main theorems to stand. Minor revisions should fix the general-hypothesis citation and the typo.","headline":"Sharp refinement of the Pintz–Révész connection for regularly and slowly varying zero-free regions, with a matching Beurling construction; solid and worth refereeing.","tokens_in":29511,"tokens_out":1542,"would_cite":true,"duration_ms":16069,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N05","11M41","11N80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A zero-free region for a Beurling zeta function forces the prime-counting error term into one explicit formula, and matching examples show the formula is sharp up to a constant.","keywords":["zero-free regions","prime number theorem with remainder","Beurling generalized number systems","Beurling zeta functions","oscillation of the error term","zero-density estimates","regularly varying functions","sharpness constructions"],"falsifier":"For $f(u)=u^{-1/2}$, the construction predicts a system whose dominant explicit-formula terms at heights $\\gamma_k=\\exp(4^k)$ add up near $x_k=u_0^{-1}(4^k)$ to oscillations of order $x_k\\exp(-\\omega(x_k)+(\\log x_k)^{1/3})$; computing these finite sums for large $k$ and checking that their signs alternate at the predicted amplitude would confirm the mechanism, while bounded or much smaller oscillations would falsify the interference claim of Theorem 1.6.","tokens_in":28508,"feed_emoji":"🔢","tokens_out":11395,"duration_ms":123503,"temperature":0.7,"pith_summary":"This paper answers, in the setting of generalized (Beurling) primes, the old question of what remainder term in the prime number theorem follows from a prescribed zero-free region. The first half proves that a zero-free region $\\sigma>1-f(\\log|t|)$, with $f$ eventually $C^1$, decreasing, strictly convex, and regularly or slowly varying, forces $\\psi_P(x)-x\\ll_\\delta x\\exp(-\\omega(x)+(A+\\delta)f(u_0(x))^B u_0(x))\\,\\omega(x)^C$ whenever the zeta function satisfies a zero-density estimate of the usual shape; here $\\omega(x)$ is the minimum of $f(u)\\log x+u$ and $u_0(x)$ is the unique minimizer. The second half constructs, for every such $f$, a Beurling number system whose zeta function has infinitely many zeros exactly on the contour $\\sigma=1-f(\\log|t|)$ and none to its right, and whose error term oscillates at the matching scale $x\\exp(-\\omega(x)+(1-\\delta)f(u_0(x))u_0(x))$. Together the two directions identify the exponential scale $x e^{-\\omega(x)}$, with a lower-order correction $f(u_0)u_0$, as the true, essentially sharp connection between zero-free regions and prime-counting error.","feed_headline":"Zero-free regions pin the prime-counting error term","feed_subtitle":"Generalized prime systems show the remainder is one exponential formula, sharp up to a constant.","key_machinery":"The carrying object is the minimizer pair $(u_0(x),\\omega(x))$: with $h(u,x)=f(u)\\log x+u$, strict convexity gives a unique minimizer $u_0(x)$, and $\\omega(x)=h(u_0(x),x)$ is the exponential decay scale of the error term. The upper-bound proof splits zeros at heights $\\gamma=\\exp(u_0(x))$, using the zero-density estimate to control how many zeros lie where the exponential weight $e^{-h(\\log\\gamma,x)}$ is non-negligible. For the reverse direction, the paper plants zeros by multiplying the zeta function by factors $G(a(s-\\rho))$ with $G(z)=(1-e^{-z}-e^{-2z})/z$, an entire function whose logarithm is the Mellin transform of a nonnegative, compactly supported function, so an infinite sequence of zeros can be introduced while keeping the generalized-prime counting function nondecreasing. The zeros are arranged in blocks near heights $\\gamma_k=\\exp(4^k)$ with real part $1-1/\\ell_k$, $\\ell_k=1/f(\\log\\gamma_k)$, so that their explicit-formula contributions interfere constructively at a selected sequence $x_k$ and destructively in the variant leading to Proposition 5.2.","core_discovery":"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.","pith_inferences":["Beyond the paper: if the same sharpness transfers to zeta functions with a fixed zero distribution, then any future narrowing of the zero-free region would automatically force a prime-counting error term of shape $x\\exp(-\\omega(x)+O(f(u_0)u_0))$; the remaining target would be removing the $\\omega(x)^C$ factor.","The pair of constructive and destructive interference constructions suggests that the error term is controlled not merely by the boundary of the zero-free region but by correlations among nearby zeros; one could test this by computing, for finite truncations of the constructed systems, how much of the oscillation survives when the zero block near $x_k$ is randomly perturbed.","A natural variational problem suggested by Theorem 1.6 is to minimize, over all Beurling systems with a fixed zero-free contour, the constant in front of $f(u_0)u_0$ in the oscillation; the construction shows the infimum is at most $1$, while the upper-bound proof only gives the constant $A$."],"forward_implications":["For the classical zeta function, the theorem applies to the asymptotically best known zero-free region, whose $f$ is regularly varying of index $-2/3$, together with existing zero-density estimates; in that situation the exponent $C$ in the density estimate matters more than the constant $A$.","For Beurling systems, the constructed examples prove that a prescribed zero-free region alone does not force any improvement over the Carlson-type zero-density estimate: one can have $N(\\sigma,T)=\\Omega(T^{(2b-1-o(1))f(\\log T)})$ while still being zero-free to the right of $1-f(\\log|t|)$.","The upper and lower bounds together imply that the epsilon in the earlier epsilonic theorems can be replaced by a function tending to zero, but not by zero: Proposition 5.2 constructs systems with zeros exactly on $\\sigma=1-(\\log|t|)^{-\\alpha}$, none to the right, and error term $O(xe^{-\\omega(x)}(\\log x)^{-1/(2(1+\\alpha))})$.","The analysis in Section 6 shows that the classical method of deducing a zero-free region from an upper bound on the zeta function is sharp, up to a universal constant, within the class of extended Beurling systems."],"supporting_citations":[{"why":"States the 1980 upper-bound answer to the classical remainder question, the epsilonic theorem this paper refines.","marker":"[18]"},{"why":"Carries the Beurling-context generalization and the explicit formula for $\\psi_P(x)$ used in the proof.","marker":"[23]"},{"why":"Gives the recent epsilon-free upper bound in the classical setting that Theorem 3.1 refines for regular or slowly varying contours.","marker":"[13]"},{"why":"Supplies the zero-planting function $G(z)=(1-e^{-z}-e^{-2z})/z$, its Mellin-transform kernel $g$, and the template for the sharpness construction.","marker":"[8]"},{"why":"Gives the Beurling zero-density estimate $N(\\sigma,T)\\ll T^{4(1-\\sigma)/(1-\\theta)}(\\log T)^9$ that instantiates the main upper bound.","marker":"[5]"},{"why":"Provides the random approximation procedure that converts the constructed extended system into a discrete Beurling number system.","marker":"[7]"},{"why":"Underpins the Section 6 discussion by giving the classical method that deduces a zero-free region from growth bounds on the zeta function.","marker":"[15]"}],"fun_headline_variants":["Zero-free regions pin the prime error term up to a constant","Beurling zeta: zero-free regions dictate error term","Sharp oscillation in prime error from zero-free contours","Prime error bound sharp up to a constant via zero-free regions","Zero-free regions bound prime error, nearly sharp"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything in the fully sharp results rests on the contour function $f$ being eventually smooth, decreasing, strictly convex, and either regularly varying of index $-\\alpha$ or slowly varying with the extra technical conditions (1.9)--(1.10); outside this class the paper does not claim the refined bounds.","fun_headline_variants_meta":{"raw":{"variants":["Zero-free regions pin the prime error term up to a constant","Beurling zeta: zero-free regions dictate error term","Sharp oscillation in prime error from zero-free contours","Prime error bound sharp up to a constant via zero-free regions","Zero-free regions bound prime error, nearly sharp"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0015,"raw_usage":{"total_tokens":6027,"prompt_tokens":966,"completion_tokens":5061,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":4981}},"tokens_in":582,"tokens_out":5061,"duration_ms":38625,"temperature":1.0,"reasoning_tokens":4981,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:18:44.540372+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $f(u)=u^{-1/2}$, the construction predicts a system whose dominant explicit-formula terms at heights $\\gamma_k=\\exp(4^k)$ add up near $x_k=u_0^{-1}(4^k)$ to oscillations of order $x_k\\exp(-\\omega(x_k)+(\\log x_k)^{1/3})$; computing these finite sums for large $k$ and checking that their signs alternate at the predicted amplitude would confirm the mechanism, while bounded or much smaller oscillations would falsify the interference claim of Theorem 1.6.","supporting_citations":[{"cited_title":"Pintz, On the remainder term of the prime number formula II","cited_arxiv_id":null,"evidence_quote":"States the 1980 upper-bound answer to the classical remainder question, the epsilonic theorem this paper refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the recent epsilon-free upper bound in the classical setting that Theorem 3.1 refines for regular or slowly varying contours."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the zero-planting function $G(z)=(1-e^{-z}-e^{-2z})/z$, its Mellin-transform kernel $g$, and the template for the sharpness construction."},{"cited_title":"Broucke, On zero-density estimates for Beurling zeta functions , to appear in Ann","cited_arxiv_id":null,"evidence_quote":"Gives the Beurling zero-density estimate $N(\\sigma,T)\\ll T^{4(1-\\sigma)/(1-\\theta)}(\\log T)^9$ that instantiates the main upper bound."},{"cited_title":"Broucke, J","cited_arxiv_id":null,"evidence_quote":"Provides the random approximation procedure that converts the constructed extended system into a discrete Beurling number system."},{"cited_title":"Landau, ¨Uber die Wurzeln der Zetafunktion , Math","cited_arxiv_id":null,"evidence_quote":"Underpins the Section 6 discussion by giving the classical method that deduces a zero-free region from growth bounds on the zeta function."}],"review_version":1}