Jia's strengthened Hardy–Littlewood conjecture for Goldbach representations

About 2 years old · traced to

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 q≤x/4q\leq x/4. There exists an absolute constant c5>0c_5>0 such that, for the even integers nn satisfying x/2<n≤xx/2<n\leq x and q∣nq\mid n,

R(n)≥c5nφ(n)⋅nlog⁡2n.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.

References

Primary source

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

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.