# Remarks on Landau–Siegel zeros

Date: Tue, Jun 18, 2024

Location: PIMS, University of British Columbia

Conference: Comparative Prime Number Theory

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory

### Abstract:

One of the central problems in comparative prime number theory involves understanding primes in

arithmetic progressions. The distribution of primes in arithmetic progressions are sensitive to real zeros near $s = 1$ of L-functions associated to primitive real Dirichlet characters. The Generalized Riemann Hypothesis implies that such L-functions have no zeros near $s = 1$. In 1935, Siegel proved the strongest known upper bound for the largest such real zero, but his result is vastly inferior to what is known unconditionally for other L-functions. We exponentially improve Siegel’s bound under a mild hypothesis that permits real zeros to lie close to $s = 1$. Our hypothesis can be verified for almost all primitive real characters. Our work extends to other families of L-functions. This is joint work with Jesse Thorner and Alexandru Zaharescu.