Generalized Limit Theorem for Mellin Transform of the Riemann Zeta-Function

Abstract

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z(s), s=σ+it, with fixed 1/2σ1, of the square |ζ(1/2+it)|2 of the Riemann zeta-function. We consider probability measures defined by means of Z(σ+iφ(t)), where φ(t), t⩾t0>0, is an increasing to +∞ differentiable function with monotonically decreasing derivative φ′(t) satisfying a certain normalizing estimate related to the mean square of the function Z(σ+iφ(t)). This allows us to extend the distribution laws for Z(s).

Keywords

• modified Mellin transform
• Riemann zeta-function
• weak convergence of probability measures