Jia's strengthened Hardy–Littlewood conjecture for Goldbach representations

Let R(n)R(n) denote the number of representations of an integer nn as a sum of two primes, and let φ\varphi denote Euler's totient function. Jia's strengthened Hardy–Littlewood conjecture. Suppose that xx is sufficiently large and qx/4q\leq x/4. There exists an absolute constant c5>0c_5>0 such that, for the even integers nn satisfying x/2<nxx/2<n\leq x and qnq\mid n,

R(n)c5nφ(n)nlog2n.R(n)\geq \frac{c_5n}{\varphi(n)}\cdot\frac{n}{\log^2 n}.

The source presents this as a stronger version of the weak Hardy–Littlewood conjecture and attributes its use in bounding Siegel zeros to Jia; the assertion itself remains unproved.

Sources & referencesView supporting material

Primary source

Yunan Wang, “Siegel Zeros and the Hardy-Littlewood Conjecture”, arXiv:2402.19332 (2024).

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