The Hardy--Littlewood asymptotic for primes p and p+4 from a uniform zeta-ratios conjecture
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 Oε(Uϑ+ε) for any fixed 1/2≤ϑ<(39−√(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 Uν, for a fixed 0<ν≤1/4. For a fixed real smooth weight supported in (1,1+1/(25h)), we obtain an error Oh,W,ε(x−Δ(ϑ)+ε) in the correlation sum with weight W(n/x)2/n, where Δ(ϑ)=7/16−ϑ+ϑ/(3−ϑ)>0. Adding the Riemann hypothesis improves this error to Oh,W,ε(xϑ−7/8+ε). 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 Bh 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 Möbius approximant in the Nyman-Beurling criterion by finite identities.
Related
- Winning property of counterexamples to Uniform Littlewood's Conjecture
- A note on the p-adic inhomogeneous Littlewood conjecture
- On the cardinality of the exception set in Littlewood's discrete conjecture
- On a question of Gowers related to Littlewood's conjecture
- The Fully Inhomogeneous p-Adic Littlewood Conjecture
BibTeX
@misc{lee2026hardylittlewood,
title = {{The Hardy--Littlewood asymptotic for primes $p$ and $p+4$ from a uniform zeta-ratios conjecture}},
author = {SeungJu Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32672},
eprint = {2610.32672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32672},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Hardy--Littlewood asymptotic for primes p and p+4 from a uniform zeta-ratios conjecture
AU - SeungJu Lee
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32672
UR - https://arxiv.org/abs/2610.32672
AB - 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_ε(U^ϑ+ε) for any fixed 1/2≤ϑ<(39−√(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^ν, for a fixed 0<ν≤1/4. For a fixed real smooth weight supported in (1,1+1/(25h)), we obtain an error O_h,W,ε(x^−Δ(ϑ)+ε) in the correlation sum with weight W(n/x)^2/n, where Δ(ϑ)=7/16−ϑ+ϑ/(3−ϑ)>0. Adding the Riemann hypothesis improves this error to O_h,W,ε(x^ϑ−7/8+ε). 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 Möbius approximant in the Nyman-Beurling criterion by finite identities.
ER -