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.