Two problems with Laurent-Stieltjes coefficients

Laurent-Stieltjes constants are the coefficients of the expansion in Laurent series of Dirichlet L-series. The interest in these constants has a long history (started by Stieltjes in 1885). Among the applications, let us cite : determining zero-free regions for Dirichlet L-functions near the real axis in the critical strip, computing the values of Riemann and Hurwitz zeta functions in the complex plane and studying the class number of the quadratic field, etc. In this book, we give explicit upper bounds for Laurent-Stieltjes constants in the following two cases: The first case when Dirichlet character is fixed and its order goes to infinity, starting from an idea due to Matsuoka for Riemann zeta function. We extend the formula of Matsuoka to Dirichlet L-functions, improving previous results. We also deduce an approximation of Dirichlet L-functions in the neighborhood of 1 by a short Taylor polynomial. The second case deals more specifically with the first Laurent-Stieltjes coefficient. We give an improvement of the known explicit upper bounds of this coefficient. This book contains new results about these constants have been published in International journals