9

The Mellin transforms of powers of *Z*(*t*)

**9.1 Introduction**

Integral transforms (Laplace, Fourier and Mellin, among others) play an important rôle in analytic number theory. Of special interest in the theory of the Riemann zeta-function ζ (*s*) are the Laplace transforms

and the (modified) Mellin transforms

where *c*(*k*) is such a constant for which the integral in (9.2) converges absolutely. The term “modified” Mellin transform seems appropriate, since customarily the Mellin transform of *f* (*x*) is defined as

Note that the lower bound of integration ...

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