On generalized shifts of the Mellin transform of the Riemann zeta-function

Abstract

In this article, we consider the asymptotic behaviour of the modified Mellin transform Z(s){\mathcal{Z}}\left(s), s=σ+its=\sigma +it, of the Riemann zeta-function using weak convergence of probability measures in the space of analytic functions defined by means of shifts Z(s+iφ(τ)){\mathcal{Z}}\left(s+i\varphi \left(\tau )), where φ(τ)\varphi \left(\tau ) is a real increasing to +∞+\infty differentiable function with monotonically decreasing derivative satisfying a certain estimate connected to the second moment of Z(s){\mathcal{Z}}\left(s). We prove in this case that the limit measure is concentrated at the point g0(s)≡0{g}_{0}\left(s)\equiv 0. This result is applied to the approximation of g0(s){g}_{0}\left(s) by shifts Z(s+iφ(τ)){\mathcal{Z}}\left(s+i\varphi \left(\tau )).

Keywords

• approximation of analytic functions
• limit theorem
• mellin transform
• riemann zeta-function
• weak convergence
• 11m06