The Okada space and vanishing of ${L(1,{f})}$

Abstract

Fix a positive integer $N \geq 2$. Following Chowla, we associate the $L$-series$L(s,f):=\sum_{n\geq 1}f(n)/n^s$ to each function $f:\mathbb{Z}\to\mathbb{C}$ with period $N$. Using a characterization derived by Okada for the vanishing of $L(1,f)$, we construct an explicit basis for the $\mathbb{Q}$-vector space, $$\mathscr{O}(N) = \{f \bmod N : f(n) \in \mathbb{Q}, L(1,f) = 0 \}.$$ We analyze the structure of this space and use the explicit basis to extend earlier works of Baker-Birch-Wirsing and Murty-Saradha.The arithmetical nature of Euler's constant $\gamma$ emerges as a central question in these extensions.