# Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series

## TL;DR

This paper explores the Laurent-Stieltjes constants related to Dirichlet L-series, providing approximations near s=1 and establishing a zero-free region for the Riemann zeta function.

## Contribution

It introduces an approximation method for Dirichlet L-functions near s=1 and proves a new zero-free region for the Riemann zeta function.

## Key findings

- Approximate Dirichlet L-functions using short Taylor polynomials near s=1
- Prove the Riemann zeta function has no zeros in |s-1| ≤ 2.2093 with 0 ≤ Re(s) ≤ 1
- Clarify properties of Laurent-Stieltjes constants for non-principal characters.

## Abstract

The Laurent Stieltjes constants $\gamma_n(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non principal, $(-1)^n\gamma_n(\chi)$ is simply the value of the $n$-th derivative of $L(s,\chi)$ at $s=1$. In this paper, we give an approximation of the Dirichlet L-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $\zeta(s)$ has no zeros in the region $|s-1|\leq 2.2093,$ with $0\leq \Re{(s)}\leq 1.$