Stelios Sachpazis: Primes in arithmetic progressions and exceptional characters

Speaker
Stelios Sachpazis, University of Turku

Abstract

Let $x\geq 1$ and let $q$ and $a$ be two coprime positive integers. As usual, $\psi(x;q,a):=\sum_{n\leq x:\,n\equiv a(\text{mod}\,q)}\Lambda(n),$ where $\Lambda$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of ”extremely” exceptional characters and established a meaningful asymptotic formula for $\psi(x;q,a)$ beyond the limitations of GRH. In particular, their asymptotic yields non-trivial information for moduli $q\leq x^{1/2+1/231}$. In this talk, we will see how one can relax the ”extremity” of the exceptional characters in their result. Then we will discuss how to improve the Friedlander-Iwaniec regime and reach the range $q\leq x^{1/2+1/82-\varepsilon}$.This talk is based on on-going work.