Monotonicity Results for Dirichlet L-functions

Abstract

We present some monotonicity results for Dirichlet $L$-functions associated to real primitive characters. We show in particular that these Dirichlet $L$-functions are far from being logarithmically completely monotonic. Also, we show that, unlike in the case of the Riemann zeta function, the problem of comparing the signs of $\frac{d^k}{d{s}^k}\log L(s,\chi )$ at any two points $s_1,s_2>1$ is more subtle.

13 pages, 2 figures, submitted for publication