The Laplace and Mellin transforms of powers of the Riemann zeta-function
Abstract
Description
This paper gives a survey of known results concerning the Laplace transform $$ L_k(s) := \int_0^\infty |ζ(1/2+ ix)|^{2k}{\rm e}^{-sx}{\rm d} x \qquad(k \in N, \R s > 0), $$ and the (modified) Mellin transform $$ {\cal Z}_k(s) := \int_1^\infty|ζ(1/2+ ix)|^{2k}x^{-s}{\rm d} x\qquad(k\in N), $$ where the integral is absolutely convergent for $\R s \ge c(k) > 1$. Also some new results on these integral transforms of $|ζ(1/2+ ix)|^{2k}$ are given, which have important connections with power moments of the Riemann zeta-function $ζ(s)$.
20 pages
20 pages