Integral and series transformations via Ramanujan's identities and\n Salem's type equivalences to the Riemann hypothesis

2012/06/06 by Semyon Yakubovich, Yakubovich, Semyon
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1206.1166

openalex publication_date 2012/06/06 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider integral and series transformations, which are associated with\nRamanujan's identities, involving various arithmetic functions and a ratio of\nproducts of Riemann's zeta functions of different arguments. Reciprocal\ninversion formulas are proved in a Banach space of functions whose Mellin's\ntransforms are integrable over the vertical line Re s > 1. Examples of new\ntransformations like Widder-Lambert and Kontorovich-Lebedev type are exhibited.\nParticular cases include familiar Lambert and Mobius transformations. Finally\na class of equivalences of the Salem type to the Riemann hypothesis is\nestablished.\n

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