Fix a positive integer $N \geq 2$. In this talk, we will focus on the problem of determining all rational valued arithmetic functions, periodic with period $N$ such that $L(1,f) := \sum_{n \geq 1} f(n)/n = 0$. This study was initiated by S. Chowla in the 1960s, drawing inspiration from Dirichlet's theorem that $L(1,\chi)\neq 0$ for a non-principal character $\chi$. We will discuss recent joint work with M. Ram Murty, wherein we use a vanishing criterion of Okada to construct an explicit basis for the $\mathbb{Q}$-vector space of functions $f \pmod N$ such that $L(1,f)=0$. This enables us to extend previous works of Baker-Birch-Wirsing and Murty-Saradha, with the arithmetic nature of Euler's constant $\gamma$ emerging as an important theme.

This talk will be accessible to graduate students.