GOLDBACH REPRESENTATIONS IN ARITHMETIC PROGRESSIONS AND ZEROS OF DIRICHLET L ‐FUNCTIONS

Abstract

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of L-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

A Survey on the Theory of Multiple Dirichlet Series with Arithmetical Coefficients as Numerators

“…Thereby we can establish the connection between Goldbach counting functions and the zeros of L-functions (especially the generalized Riemann hypothesis, GRH). The existence of this connection was first suggested by Granville [20,21] in the Riemann zeta case and then fully developed in Bhowmik et al [3,4,5]. We conclude this paper with the statement of some theorems proved in those papers.…”

Average Goldbach and the Quasi-Riemann Hypothesis

“…2. In general we are yet unable to establish an equivalence between average orders of restricted Goldbach functions and appropriate quasi Riemann Hypotheses for Lfunctions of [4] or [1] though a particular case is handled in [1] using the method of this note.…”

“…Using a better kernel and a more careful calculation we can improve the bound to δ ′ = δ/3. It is of little importance in this context as the methods of [1,2,4] give δ ′ = δ as soon as we have δ ′ < 1, through an explicit expression of S(x) using roots of the zeta-function. We are unable to get δ ′ = δ directly by the simple method of this paper and we think it cannot be done.…”

“…a good asymptotic order of the Goldbach function is indeed equivalent to the Riemann Hypothesis. The same idea has since been applied to the restricted sums of two primes both in the same congruence class (Section 8 [1]).…”

On an Average Goldbach Representation Formula of Fujii

“…Weaker results of this type were first obtained by Granville [Gra07] [Gra08]. To prove this theorem we only need to adjust a few details of the proof in [BHMS19] to match our earlier theorems. With more work the power of log N can be improved but that makes no difference in applications.…”