Stelios Sachpazis: Primes in arithmetic progressions and exceptional characters

Abstract:

Let \(x \ge 1\) 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.