New summation and transformation formulas of the Poisson, M "untz,\n M "obius and Voronoi type

2014/10/29 by Semyon Yakubovich, Yakubovich, Semyon
Mathematics · Physics and Astronomy · #11M06 #11M36 #11N 37 #33C10 #44A15 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.1410.7934

openalex publication_date 2014/10/29 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Basing on properties of the Mellin transform and Ramanujan's identities,\nwhich represent a ratio of products of Riemann's zeta- functions of different\narguments in terms of the Dirichlet series of arithmetic functions, we obtain a\nnumber of the Poisson, M "untz, M "obius and Voronoi type summation\nformulas. The corresponding analogs of the M "untz operators are\ninvestigated. Interesting and curious particular cases of summation formulas\ninvolving arithmetic functions are exhibited. Necessary and sufficient\nconditions for the validity of the Riemann hypothesis are derived.\n

Related

• On Müntz-type formulas related to the Riemann zeta function
• Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities
• Summation rules for the values of the Riemann zeta-function and generalized harmonic numbers obtained using Laurent developments of polygamma functions and their products
• Summation and the Poisson formula
• Integral and series transformations via Ramanujan's identities and\n Salem's type equivalences to the Riemann hypothesis