Hall 123

Chowla's problem and its generalizations

Siddhi Pathak

Queen's University

Inspired by the mystery surrounding Dirichlet's theorem about non-vanishing of L(1,\chi),for a non-principal Dirichlet character \chi, Chowla asked the following question in early 1960's:

Let $f$ be a rational-valued arithmetical function that is periodic with prime period $p$, not identically zero. Then is it true that
\sum_n f(n)/n
eq 0?

This question was resolved in the case of odd functions, ie, f(p-n) = -f(n), by Chowla himself. In 1972, Baker, Birch and Wirsing answered
Chowla's question in a much more general setup.

We will discuss an application of Bass's theorem to the characterisation of algebraic-valued, periodic functions $f$ for which the above series vanishes. We will also introduce the Lerch zeta function and mention a few results about the transcendence of its certain special values. We will define a generalised series attached to a periodic function and obtain a similar non-vanishing result. This is joint work with Prof. Ram Murty.