The Hardy--Littlewood asymptotic for primes $p$ and $p+4$ from a uniform zeta-ratios conjecture

Source abstract

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 Möbius approximant in the Nyman--Beurling criterion by finite identities.