## The Chowla Problem and its generalizations

##### Abstract

In this thesis, we study the vanishing of certain $L$-series attached to periodic arithmetical functions. Throughout the writeup, we let $f$ be an algebraic-valued (at times even rational-valued) arithmetical function, periodic with period $q$. Define
\begin{equation*}
L(s,f) := \sum_{n=1}^{\infty} \frac{f(n)}{n^s}.
\end{equation*}
We discuss a conjecture of Sarvadaman Chowla, made in the early 1960's of the non-vanishing of $L(1,f)$ for rational-valued functions $f$. We further discuss the theorem of Baker-Birch-Wirsing that classifies all odd algebraic-valued, periodic arithmetical functions $f$ with $L(1,f)=0$. We then apply a beautiful result of Bass to give a necessary condition for even algebraic-valued, periodic arithmetical functions $f$, to satisfy $L(1,f)=0$. The sufficiency condition obtained via this method is unfortunately not very clean. This, in view of a theorem of Ram Murty and Tapas Chatterjee, completes the characterization of all algebraic-valued periodic arithmetical functions $f$ with $L(1,f) = 0$.
In the next part of the thesis, we discuss the Lerch-zeta function and its functional equation. We also deduce the transcendence of its certain special values. Keeping this in mind, we define a new $L$-series attached to periodic functions and deduce a necessary condition for the vanishing of this $L$-series at $s=1$.
Finally, we mention some research topics that were stumbled upon during the course of this study and a few partial results regarding these questions.