A Class of Dirichlet Series Integrals

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 e...

Special values of trigonometric Dirichlet series and Eichler integrals

We provide a general theorem for evaluating trigonometric Dirichlet series of the form ∑ n>1 f(πnτ) ns , where f is an arbitrary product of the elementary trigonometric functions, τ a real quadratic irrationality and s an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating th...

Exact Estimates for Integrals Involving Dirichlet Series with Nonnegative Coefficients

We consider the Dirichlet series ∞ ∑ k=2 akk −1−x =: f(x), x > 0, with coefficients ak ≥ 0 for all k. Among others, we prove exact estimates of certain weighted Lp-norms of f on the unit interval (0, 1) for any 0 < p <∞, in terms of the coefficients ak . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for int...

On a Secant Dirichlet Series and Eichler Integrals of Eisenstein Series

We consider, for even s, the secant Dirichlet series ψs(τ) = ∑∞ n=1 sec(πnτ) ns , recently introduced and studied by Laĺın, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Laĺın, Rodrigue and Rogers, that the values ψ2m( √ r), with r > 0 rational, are rational multiples of π2m. We then put the properties of the secant Dirichlet series into context by showing ...

On the Selberg Class of Dirichlet Series: Small Degrees