New summation and transformation formulas of the Poisson, M\"{u}ntz, M\"{o}bius and Voronoi type

TL;DR

This paper derives new summation and transformation formulas related to the Poisson, M"{u}ntz, M"{o}bius, and Voronoi types using Mellin transforms and Ramanujan's identities, with implications for the Riemann hypothesis.

Contribution

It introduces novel summation formulas and analogs of M"{u}ntz operators, expanding the analytical tools for studying zeta functions and the Riemann hypothesis.

Findings

New summation formulas for arithmetic functions

Analogs of M"{u}ntz operators analyzed

Conditions for Riemann hypothesis validity derived

Abstract

Basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta- functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of the Poisson, M\"{u}ntz, M\"{o}bius and Voronoi type summation formulas. The corresponding analogs of the M\"{u}ntz operators are investigated. Interesting and curious particular cases of summation formulas involving arithmetic functions are exhibited. Necessary and sufficient conditions for the validity of the Riemann hypothesis are derived.