On an Average Goldbach Representation Formula of Fujii

Abstract

Fujii obtained a formula for the average number of Goldbach representations with lower order terms expressed as a sum over the zeros of the Riemann zeta-function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result, and obtain applications conditional on various conjectures on zeros of the Riemann zeta-function.

On an Average Goldbach Representation Formula of Fujii

“…The second equation actually depends on RH, which is included from our assumption on L(s, χ 0 ). The method we use is a combination of the methods of the papers [GS23] and [FGIS22]. We can use this method to prove (5) but the proof is much more complicated than the proof of Theorem 1.…”

The Prime Number Theorem and Pair Correlation of Zeros of the Riemann Zeta-Function

“…Recently we obtained a quantitative improvement in the bound for R(x) as an application of a method related to a formula of Fujii [GS21]. The main result we obtained is that, assuming RH and…”

An explicit Selberg mean-value result with applications

“…1]. Other interval results for Goldbach numbers include those under certain assumptions, such as in Corollary 3 of [8] for intervals (x, x + C log x), and estimates on the average number of Goldbach numbers in intervals [9]. We prove the following explicit version of Montgomery and Vaughan's result in Section 4.…”