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