Fix a positive integer N >= 2. Following Chowla, we associate the L-series L(s, f) := Sigma(n >= 1) f(n)/n(s) to each function f : Z -> C with period N. Using a characterization derived by Okada for the vanishing of L(1, f), we construct an explicit basis for the Q-vector space, partial derivative (N) = {f mod N : f(n) is an element of 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.
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机译：修复一个正整数N > = 2。我们把L系列L (s、f): =σ(n > =1) f (n) / n (s)到每个函数f: Z - > C期使用特性得到了n冈田克也消失的L (f),我们构造Q-vector空间,一个明确的基础部分导数(N) = {mod N: f (N)是一个元素Q L (f) = 0}。空间和使用显式的基础上扩展早些时候Baker-Birch-Wirsing和工作Murty-Saradha。常数伽马出现作为一个核心问题这些扩展。
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