The Critical Values of Certain Dirichlet Series

We investigate the values of several types of Dirichlet series D(s) for certain integer values of s, and give explicit formulas for the value D(s) in many cases. The easiest types of D are Dirichlet L-functions and their variations; a somewhat more complex case involves elliptic functions. There is one new type that includes ∑∞ n=1(n +1) for which such values have not been studied previously. 2000 Mathematics Subject Classification: 11B68, 11M06, 30B50, 33E05. Introduction By a Dirichlet character modulo a positive integer d we mean as usual a C-valued function χ on Z such that χ(x) = 0 if x is not prime to d, and χ induces a character on (Z/dZ). In this paper we always assume that χ is primitive and nontrivial, and so d > 1. For such a χ we put (0.1) L(s, χ) = ∞ ∑ n=1 χ(n)n. It is well known that if k is a positive integer such that χ(−1) = (−1), then L(k, χ) is π times an algebraic number, or equivalently, L(1 − k, χ) is an algebraic number. In fact, there is a well-known formula, first proved by Hecke in [3]: (0.2) kdL(1− k, χ) = − d−1 ∑ a=1 χ(a)Bk(a/d), where Bk(t) is the Bernoulli polynomial of degree k. Actually Hecke gave the result in terms of L(k, χ), but here we state it in the above form. Hecke’s proof is based on a classical formula (0.3) Bk(t) = −k!(2πi) −k ∑ 06=h∈Z he(ht) (0 < k ∈ Z, 0 < t < 1). Documenta Mathematica 13 (2008) 775–794