The Laplace and Mellin transforms of powers of the Riemann zeta-function

TL;DR

This paper surveys known results and presents new findings on the Laplace and Mellin transforms of powers of the Riemann zeta-function, highlighting their connections to the zeta-function's power moments.

Contribution

It provides a comprehensive survey and introduces new results on integral transforms of zeta-function powers, linking them to moment problems.

Findings

New results on integral transforms of |zeta(1/2+ ix)|^{2k}

Connections established between transforms and zeta-function moments

Enhanced understanding of the analytic properties of these transforms

Abstract

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