Explicit bounds on exceptional zeroes of Dirichlet $L$-functions

TL;DR

This paper improves explicit bounds on the exceptional zeroes of Dirichlet L-functions by refining estimates of their derivatives near the critical line, enhancing understanding of their distribution.

Contribution

It introduces improved explicit bounds for the exceptional zeroes of Dirichlet L-functions through refined estimates of L'(\sigma, \chi) near \sigma=1.

Findings

Tighter bounds on exceptional zeroes of Dirichlet L-functions.

Enhanced explicit estimates for L'(\sigma, \chi) near \sigma=1.

Improved understanding of zero distribution near the critical line.

Abstract

The aim of this paper is to improve the upper bound for the exceptional zeroes $\beta_0$ of Dirichlet $L$-functions. We do this by improving on explicit estimate for $L'(\sigma, \chi)$ for $\sigma$ close to unity.