Van der Pol Expansions of L-Series

We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series. 1. Preliminaries. In [8] the following odd looking integral evaluation is obtained. ∫ ∞ 0 ( 3− 2 √ 2 cos (t log 2) ) |ζ (1/2 + it)| t2 + 1/4 dt = π log 2. (1) This identity turns out—formally—to be a case of a rather pretty, and perhaps useful, class of L-series evaluations given in Theorem 1 and Corollary 1 (cf. [2]). In Theorem 3 we recover (1) entirely rigorously. Given a Dirichlet series λ(s) := ∞ ∑ n=1 λn ns , s = σ + iτ, σ = 0, we consider the integral ιλ(σ) := 1 2 ∫ ∞ −∞ ∣∣∣∣ λ(s) s ∣∣∣∣ 2 dτ as a function of λ. Observe that when the coefficients λn are real ιλ(σ) = ∫ ∞ 0 ∣∣∣∣ λ(s) s ∣∣∣∣ 2 dτ, but that this is not necessarily so when the coefficients are complex. We refer to [3, 5, 7] for other, largely standard details. 2. Integrals Involving s with Large RealPart. It is convenient to recall [2] that, for u, a > 0, ∫ ∞ 0 cos (at) t2 + u2 dt = π 2u e−au. (2) *Department of Mathematics, University of Western Ontario, London, Ontario, N6A 5B7. Research supported by NSERC. E-mail: [email protected] †Faculty of Computer Science, Dalhousie University, Halifax NS, B3H 1W5. Research supported by NSERC, the Canada Foundation for Innovation and the Canada Research Chair Program. E-mail: [email protected] 2000 Mathematics Subject Classification. 11M35, 11M41, 30B50.