A Class of Dirichlet Series Integrals

In this note we extend the solution to a recent Monthly problem to analyze a broad class of Dirichlet series and illustrate the result in action in various ways. More precisely, in [6] the following integral evaluation is obtained: ∞ 0 3 − 2 √ 2 cos (t log 2) |ζ (1/2 + it)| 2 t 2 + 1/4 dt = π log 2. (1) This somewhat recondite-looking result transpires to be a case of a rather pretty class of evaluations given in Theorem 1, and worth making explicit. We let λ(s) := n>0 λ n n −s represent a formal Dirichlet series, with real coefficients λ n and we set s := σ + i τ with σ = (s) > 0. We refer to [2], [3], and [7] for other, largely standard, notation. We consider the following integral ι λ (σ) := ∞ 0 λ(s) s 2 dτ, (2) as a function of the Dirichlet series λ (i.e., of the coefficients λ n). The question of evaluating (2) when λ is the alternating zeta-function was posed and solved (at least formally) in [6]. It is convenient to know that for strictly positive u and a ∞ 0 cos (at) t 2 + u 2 dt = π 2u e −au , (3) as may be established by contour integration, by consulting a computer algebra system, or otherwise. 2. The Main Result. Theorem 1. For λ(s) = ∞ n=1 λ n n −s and s = σ + i τ with fixed σ = (s) > 0 such that the Dirichlet series is absolutely convergent it is true that ι λ (σ) = ∞ 0 λ(s) s 2 dτ = π 2σ ∞ n=1 Λ 2 n − Λ 2 n−1 n 2σ , (4) where Λ n := n k=1 λ k and Λ 0 := 0.