The Hardy-Littlewood asymptotic for prime pairs of fixed even difference from a uniform zeta-ratios conjecture

Seungju Lee

REVIEW

The Hardy-Littlewood asymptotic for prime pairs of fixed even difference from a uniform zeta-ratios conjecture

SeungJu Lee

Number Theory · Submitted 23 Oct 2026

Pith's verdict: Serious conditional result on prime pairs of gap four; the diagonal-term factor error flagged by the stress test is real but repairable, and the central mechanism survives.

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arXiv ID2610.32672 (version 2)

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CitationSeungju Lee. "The Hardy-Littlewood asymptotic for prime pairs of fixed even difference from a uniform zeta-ratios conjecture." arXiv:2610.32672 (2026). https://pith.science/paper/IB7OQRJP

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Editor's note

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Read this one. The paper does something genuinely new: from a restricted two-over-two zeta-ratios formula it derives the Hardy–Littlewood asymptotic for primes p and p+4 with a quantitative error, without assuming GRH, and it improves the error under RH. The differentiated ratios mean and the retention of crossed zero residues are the real innovations, and the singular series S(4) emerges from a Dirichlet series rather than being inserted. The conditional theorems are honestly stated, and the citation pattern is clean.

The stress-test note gets part of the diagnosis right but misses half the arithmetic. In Lemma 3.1, the diagonal contribution to |P_x(t)|^2 is T/2 times H(0) times the sum, not T^2 H(0) times the sum: g is even, so the integral from 0 to infinity of g(t/T) is T/2 times its Fourier transform at zero. The stress-test note's proposed correction T H(0) is also off by a factor of two. The same factor error appears in Proposition 4.2, where the line immediately before the display gives TG(1) = T phi(0)/2, and then the next display says T^2 phi(0). So the written equations are false as printed, but the error is consistent and has the shape of a propagating typo. Redefine D_x as (T/2)H(0) times the sum, and the cancellation in Proposition 6.2 goes through unchanged; the final correlation estimate and the main theorem are not affected. This is a referee request, not a refutation.

The larger caveat is structural: the entire tower rests on Hypothesis 2.1, a very strong uniform ratios conjecture with error O(U^{1/2+eps}) for shifts whose imaginary parts are allowed to grow like U^{1/4}. The paper is honest that every theorem is conditional, but a reader should not underestimate how strong this input is. The closing declaration that GPT was used in generating proofs matters: the diagonal factor slip is exactly the kind of error that needs a human pass, so independent verification should be required before the technical chain is trusted. Section 8, on Nyman–Beurling approximants, is mostly independent and adds little to the main result.

If I were an editor, I would send this to a number theorist who knows the ratios conjectures. It deserves a serious referee rather than a desk rejection, and after the diagonal correction and a careful independent check, the conditional theorem may well stand. I would not cite the current version until the factor error is fixed and the GPT-generated parts have been verified.

Referee report

Significance

If the proof were correct, this would be a notable conditional result: the full prime-pair asymptotic for a fixed gap is obtained from a ratios conjecture without assuming RH, the singular-series constant 2C_2 emerges from the computation rather than being inserted, and the error term is quantitative. The treatment of the crossed zero residues and the two-weight bracket is an interesting device, and the supporting lemmas (Mellin truncations, zero-count bounds, singular-series inversion) are plausible in outline. However, two load-bearing algebraic inconsistencies in the displayed formulas (the diagonal term in Sections 3-5 and the definition of Y in Section 2) mean that, as submitted, the arguments do not currently establish the theorems; the paper is not yet in publishable form.

Major objections (2)

[Section 3, Eq. (3.9); Lemma 3.1; Propositions 4.2 and 5.3] The diagonal term contains a factor-of-T error. Since g_+ and g_- are even, int_0^infty g_+(v) dv = phi_+(0)/2 = H(0)/2, so the n=m contribution to I_+ is (T/2)H(0) sum_n Lambda(n)^2 n^{-1} W(n/x)^2, not T^2 H(0) sum_n ... as in (3.9). The same erroneous diagonal is asserted in (4.9) and (5.7), and in the proof of Proposition 4.2 the displayed equality T G(1) sum ... = T^2 phi(0) sum ... is algebraically false because G(1)=phi(0)/2. Consequently the cancellation in Proposition 6.2 subtracts a term that is not present: with the printed D_x, the two sides of the inequality differ by (T^2 - T/2)H(0) sum_n Lambda(n)^2 n^{-1} W(n/x)^2, which is O(x^2 log^2 x) and, after division by T, dominates the claimed error O(x^{-1/16+eps}). The proof structure can be repaired by replacing T^2 H(0) by T H(0)/2 consistently, but as written (6.4), Theorem 1.2, and Theorem 1.1 are not established.
[Section 2, Eq. (2.2), proof of Lemma 2.2] The displayed definition of Y does not have the property used in the proof. With beta=gamma=delta=eta, direct substitution in (2.1)-(2.2) gives A(alpha,eta;eta,eta)=1 and Y(alpha,eta;eta,eta)=zeta(1+2eta)^2 zeta(1+alpha+eta)^2, not 1, and Y(-eta,-alpha;eta,eta) does not vanish but contains zeta(1) factors. Hence the asserted cancellation leading to int_0^U zeta(1/2+alpha+it)/zeta(sigma_0+it) dt = U + O(U^{1/2+eps}) does not follow from the displayed formula. Since this is the basis for the zero-free strip (2.6) and the zero-count bound (2.8), and those are used in Lemma 6.1, the printed proofs of Lemmas 2.2 and 2.3 are not valid as they stand. The intended Y is presumably the standard ratios-theory expression with zeta factors in the denominator; that definition must be corrected and the subsequent identities rechecked.

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The paper

The paper claims

This paper proves the Hardy-Littlewood asymptotic for prime pairs of difference four from one restricted zeta-ratios formula, without the Riemann hypothesis.

The reading

This paper shows that the Hardy-Littlewood prediction for prime pairs of distance four — the claim that the number of primes $p\le x$ with $p+4$ also prime behaves like $2C_2x/(\log x)^2$ — follows from a single restricted version of the L-functions ratios conjecture, an assumed formula for averages of ratios of zeta values. That formula, Hypothesis 2.1, is the paper's only unproved input: the Riemann hypothesis is not assumed, and even the zero-free strip and zero-count bounds used in the proof are extracted from the formula itself rather than from standard theorems. The argument keeps the crossed zero residues in a Mellin inversion and uses two nonnegative weights whose Fourier transforms agree at zero but are equal and opposite at the frequency that selects the shift four, so the same prime-pair correlation is squeezed between two bounds that agree to the stated error. If the paper is right, the case $k=4$ of Conjecture B is a corollary of one statistical hypothesis about $\zeta$, with a smoothed correlation error $O_\varepsilon(x^{-1/16+\varepsilon})$ that improves to $O_\varepsilon(x^{-3/8+\varepsilon})$ under the Riemann hypothesis.

What carries the argument

The argument is carried by four objects. First, Hypothesis 2.1, the uniform two-over-two zeta-ratios formula (2.3): an integral average of a ratio of four zeta values over a segment of the critical line, asserted to equal an explicit main term plus a swapped-term integral with error $O_\varepsilon(U^{1/2+\varepsilon})$; from this one formula the paper derives the zero-free strip $35/64\le\Re s\le37/64$, the bound $\zeta'/\zeta(9/16+iv)\ll\log^2(2+|v|)$, and the zero count $N_{>}(9/16,T)\ll_\varepsilon T^{1/2+\varepsilon}$ (Lemma 2.2). Second, the two nonnegative time weights $g_\pm(t)=h(t)^2(1\pm\cos(8\pi t))$, whose Fourier transforms $\phi_\pm$ agree at zero and satisfy $\phi_-=-\phi_+$ on the side intervals that select the shift $j=4$; the two inequalities of (6.5), each bounding $I_\pm(x)$ from below by the $|Q_x|^2$ moment minus crossed pole-zero error terms, pinch the difference-four correlation between matching upper and lower bounds. Third, the Mellin decomposition $P_x(t)=Q_x(t)+R_x(t)-Z_x(t)$, which keeps the crossed zero residues $Z_x$ in play and bounds their mixed term with $Q_x$ using the zero-count estimates. Fourth, the singular-series Dirichlet series $D_S(z)=\sum_k S(k)k^{-z}=\zeta(z)\zeta(1+z)A(z)$, whose Fourier inversion collapses the main terms exactly to $S(4)=2C_2$. Section 8 adds the Nyman-Beurling approximant $a_N(d)=\mu(d)(1-\log d/\log N)$; through it the truncated Chebyshev function is expressed and the exact second-difference identity (8.7) is obtained.

Extended reading notes

Core claim

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.

Load-bearing premise

Every theorem rests on Hypothesis 2.1, the unproved assertion that an average of a specific ratio of four zeta-function values over an interval of length $U$ equals its two explicit main terms with error $O_\varepsilon(U^{1/2+\varepsilon})$, uniformly over the stated box of shifts; if that formula fails, the zero-free strip, the zero count, the differentiated ratio mean, and the prime-pair asymptotics all collapse with it.

Editorial extensions

If this is right

The full Hardy-Littlewood asymptotic for difference four — Conjecture B at $k=4$ — is a corollary of the restricted ratios formula (2.3), with the Riemann hypothesis playing no role in the main theorem.
For fixed smooth weights the correlation error is $O_\varepsilon(x^{-1/16+\varepsilon})$ unconditionally and $O_\varepsilon(x^{-3/8+\varepsilon})$ under RH, so a qualitative conjecture becomes quantitative control on weighted prime-pair sums.
The logarithmically weighted sum $B_4(x)=2C_2\log x+o(\log x)$ and the second-difference asymptotic $F_x(5)-2F_x(4)+F_x(3)=4C_2x+o(x)$ follow, the latter connecting the pair problem to the Nyman-Beurling criterion.
Because $S(4)=2C_2>0$, the results give eventual lower bounds: $B_4(x)\ge c\log x$ for every $c<2C_2$, and the relevant second difference is at least $cx$ for every $c<4C_2$, for all sufficiently large $x$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The pinching mechanism is not visibly tied to the number four: the same scheme would handle any fixed even difference $2k$ provided one can construct nonnegative weights whose Fourier transforms agree at zero and are opposite in sign on the side interval selecting that $k$. The paper does not claim this; it is the natural next test of the method.
The unconditional error exponent $-1/16$ and the RH exponent $-3/8$ are set by the zero-count bounds extracted from Hypothesis 2.1, so any future strengthening of those bounds would feed directly into the correlation error; the proof shows this coupling but does not state it as a general principle.
Lemma 2.2 turns a statistical averaging assumption into the concrete arithmetic fact that $\zeta$ has no zeros with real part in $[35/64,37/64]$. Because this is checkable with existing zero-location algorithms, the hypothesis carries a sharp, falsifiable fingerprint that deserves direct computational scrutiny.
If the same pinching argument could be run for every even difference at once under one uniform ratios formula, the entire Hardy-Littlewood Conjecture B would follow by the same mechanism; this paper proves only the case $k=4$, so the general claim remains an extrapolation.

Original abstract

We prove the Hardy-Littlewood asymptotic for prime pairs of every fixed positive even difference $h$ from a restricted uniform two-over-two zeta-ratios formula. The formula may have error $O_\varepsilon(U^{\vartheta+\varepsilon})$ for any fixed $1/2\le\vartheta<(39-\sqrt{177})/32$. The real parts of all four shifts lie in $[3/64,5/64]$; their imaginary parts need only have absolute value at most $U^\nu$, for a fixed $0<\nu\le1/4$. For a fixed real smooth weight supported in $(1,1+1/(25h))$, we obtain an error $O_{h,W,\varepsilon}(x^{-\Delta(\vartheta)+\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$, where $\Delta(\vartheta)=7/16-\vartheta+\vartheta/(3-\vartheta)>0$. Adding the Riemann hypothesis improves this error to $O_{h,W,\varepsilon}(x^{\vartheta-7/8+\varepsilon})$. The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral. Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference $h$. We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_h$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function. The latter is connected to a logarithmically weighted M\"obius approximant in the Nyman-Beurling criterion by finite identities.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-26: 1 request, 6,447 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.59 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.27 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We prove the Hardy--Littlewood asymptotic for prime pairs of difference four from a restricted uniform two-over-two zeta-ratios formula. The formula has error $O_\varepsilon(U^{1/2+\varepsilon})$, real parts of all four shifts in $[3/64,5/64]$, and imaginary parts of absolute value at most $U^{1/4}$. For a fixed real smooth weight supported in $(1,101/100)$, we obtain an error $O_\varepsilon(x^{-1/16+\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$. Adding the Riemann hypothesis improves this error to $O_\varepsilon(x^{-3/8+\varepsilon})$. The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral. Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference four. We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_4$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function. The latter is connected to a logarithmically weighted M\"obius approximant in the Nyman--Beurling criterion by finite identities.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove the Hardy--Littlewood asymptotic for prime pairs of difference four from a restricted uniform two-over-two zeta-ratios formula.	result 0.61	0.96	0.95	asserted 1.00
s1	The formula has error $O_\varepsilon(U^{1/2+\varepsilon})$, real parts of all four shifts in $[3/64,5/64]$, and imaginary parts of absolute value at most $U^{1/4}$.	result 0.63	0.66	0.05	asserted 0.99
s2	For a fixed real smooth weight supported in $(1,101/100)$, we obtain an error $O_\varepsilon(x^{-1/16+\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$.	result 0.97	0.93	0.06	asserted 1.00
s3	Adding the Riemann hypothesis improves this error to $O_\varepsilon(x^{-3/8+\varepsilon})$.	result 0.96	0.77	0.06	asserted 0.98
s4	The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral.	method 0.99	0.91	0.03	asserted 1.00
s5	Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference four.	result 0.58	0.67	0.03	asserted 1.00
s6	We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_4$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function.	result 0.98	0.94	0.19	asserted 1.00
s7	The latter is connected to a logarithmically weighted M\"obius approximant in the Nyman--Beurling criterion by finite identities.	interpretation 0.72	0.78	0.04	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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