**Date & Time:** Friday, June 07, 2013, 3.30 p.m

**Venue:**Ramanujan Hall

**Title:** Ramanujan-Hardy-Littlewood-Riesz phenomena and monotonicity results for
Dirichlet L-functions

**Speaker:** Atul Dixit, Tulane University

**Abstract:** We generalize a result of Ramanujan, corrected by Hardy and Littlewood,
to primitive Hecke forms, which interestingly exhibits faster
convergence than in the initial case of the Riemann zeta function. We
also provide a criterion in the spirit of Riesz for the Riemann
Hypothesis for the associated L-functions. We will briefly mention the
corresponding results for Dirichlet L-functions.
We also 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. If time permits, we will also 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.

This is joint work with Arindam Roy and Alexandru Zaharescu.