The Mellin transform and the Riemann zeta-function
by
Aleksandar Ivic
University of Belgrade

The Mellin transform of \Cal M [f(x)] of a function f(x) is

\Cal M [f(x)] = ó õ \infty 0  f(x)xs-1 dx        (s=\sigma+it).

This integral transform is of fundamental importance in Analytic Number Theory. Some applications to the Riemann zeta-function \zeta(s) will be discussed. The first involves \Delta_k(x), the error term in the asymptotic formula for the summatory function of d_k(n), the arithmetic function generated by \zeta^k(s), k in N. The second one is the function

\Cal Zk(s) : = ó õ \infty 1  |\zeta(\frac 12 +ix)|2kx-sdx     (k in N,  , \sigma >= \sigma0(k) > 1),

in particular the important cases k=1, 2. Some recent results on \Cal Z_k(s), obtained in a joint work with M. Jutila and Y. Motohashi, will be presented.

Date received: January 2, 1999