Arithmetic and analytic aspects of values of L-functions [HBNI Th240]

Abstract

The central theme of this thesis is to study some analytic and arithmetic properties of values of L-functions at “special points”. The values of L-functions encode a lot of arithmetic data and are at the heart of several deep mysteries. The Riemann hypothesis which predicts that all non-trivial zeros of the Riemann zeta function lie on the line ℜ(s) = 1/2 is one such enigma. For a non-trivial Dirichlet character χ, it is expected that L(s, χ) does not vanish at 1/2. Though this problem is still wide open, a lot of progress has been made in recent years.