Primes in arithmetic progressions under the presence of Landau-Siegel zeroes

TL;DR

This paper extends previous results on primes in arithmetic progressions by relaxing conditions on Landau-Siegel zeroes, providing a broader asymptotic formula for the von Mangoldt sum in this context.

Contribution

It significantly relaxes the extremity conditions on Landau-Siegel zeroes needed for asymptotic formulas in prime number theory.

Findings

Conditional asymptotic formula for $\psi(x;q,a)$ in wider range of $q$

Relaxed assumptions on Landau-Siegel zeroes

Extended range of moduli for prime distribution analysis

Abstract

Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $\psi(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}\Lambda(n)$, where $\Lambda$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of exceptional characters corresponding to "extreme" Landau-Siegel zeroes and established a meaningful asymptotic formula for $\psi(x;q,a)$ beyond the square-root barrier of the Generalized Riemann Hypothesis. In particular, their asymptotic yields non-trivial information for moduli $q\leqslant x^{1/2+1/231}$. In this paper, we considerably relax the extremity of the Landau-Siegel zero required in the work of Friedlander and Iwaniec and obtain a conditional asymptotic formula for $\psi(x;q,a)$ in a slightly wider range of $q$.