On two problems concerning the Laurent Stieltjes coefficients of Dirichlet L-series

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 a non-principal, (-1)^n \gamma_n(\chi) is simply
the value of the n-th derivative of L(z, \chi) at z=1.

The interest in these constants has a long history (going back to
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 0<=\Re(z)<= 1,
-computing the values of the Riemann and Hurwitz zeta functions in the
complex plane and
-studying the class number of the quadratic field, etc.

In this talk, I will give explicit upper bounds for the Laurent-Stieltjes
constants in the following two cases:

-The character \chi is fixed and the order n goes to infinity.
-The order n is 0 and the modulus q goes to infinity.