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

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 $0\leq \Re{(s)}\leq 1.$