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

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

2006-05-29

arxiv.org/abs/math/0605721v2

International J. of Mathematics and Analysis Vol. 1 No. 2, 2006, pp. 131-140

Mathematics

Number Theory

20 pages

Scientific paper

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 $\zeta(s)$.

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