Monotonicity Results for Dirichlet L-functions

TL;DR

This paper investigates the monotonicity properties of Dirichlet L-functions associated with real primitive characters, revealing they are not logarithmically completely monotonic and highlighting the complexity of sign comparisons of derivatives.

Contribution

The paper provides new monotonicity results for Dirichlet L-functions and demonstrates their differences from the Riemann zeta function regarding derivative sign comparisons.

Findings

Dirichlet L-functions are not logarithmically completely monotonic.

Sign comparison of derivatives of log L-functions is more complex than for the Riemann zeta function.

Monotonicity properties vary depending on the character and the point in the domain.

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 $\frac{d^k}{ds^k}\log L(s,\chi)$ at any two points $s_1, s_2>1$ is more subtle.