The Laplace And Mellin Transforms Of Powers Of The Riemann Zeta Function

Abstract

This paper gives a survey of known results concerning the Laplace transform Lk(s) := Z Â¥ 1 jz ( 1 2 +ix)j2keð€€€sx dx (k 2 N, Âe(s) > 0 ) and the (modified) Mellin transform Zk(s) := Z Â¥ 1 jz ( 1 2 +ix)j2kxð€€€s dx where the integral is absolutely convergent for Âe(s) > c(k) Equations involving the Mellin and Laplace transform of functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions to differential and integral equations. In this work we trace a brief history of the development of the Riemann Zeta function,Laplace transform and Mellin transforms of functions, illustrate the close relationship between them and present range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Our literature review will show that while continued research into Mellin properties, identities generated many new results, some of these can be shown to be variations of known identities.