The Hardy-Littlewood asymptotic for prime pairs of fixed even difference from a uniform zeta-ratios conjecture
Year: 2026
Type: preprint
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 $U^ν$, for a fixed $0<ν\le1/4$. For a fixed real smooth weight supported in $(1,1+1/(25h))$, we obtain an error $O_{h,W,\varepsilon}(x^{-Δ(\vartheta)+\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$, where $Δ(\vartheta)=7/16-\vartheta+\vartheta/(3-\vartheta)>0$. 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 Möbius approximant in the Nyman-Beurling criterion by finite identities.
Source: arXiv (Cornell University)
Author: Lee, SeungJu
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223693438