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arXiv · 1410.7934

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

Semyon Yakubovich
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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.

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Semyon Yakubovich. 2014-10-29. New summation and transformation formulas of the Poisson, M\"{u}ntz, M\"{o}bius and Voronoi type. https://arxiv.org/abs/1410.7934

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