Hypothesis
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On the Riemann-Hardy Conjecture for the Ramanujan Zeta-Function
Version 1
: Received: 29 May 2021 / Approved: 31 May 2021 / Online: 31 May 2021 (12:34:12 CEST)
How to cite: Yang, X. On the Riemann-Hardy Conjecture for the Ramanujan Zeta-Function. Preprints 2021, 2021050769. https://doi.org/10.20944/preprints202105.0769.v1 Yang, X. On the Riemann-Hardy Conjecture for the Ramanujan Zeta-Function. Preprints 2021, 2021050769. https://doi.org/10.20944/preprints202105.0769.v1
Abstract
In this article we propose the integral, series and product representations for the Ramanujan zeta-function. We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function. We present some variants for the product for the Ramanujan $\Xi$-function. We prove that all of its zeros are real. Along the way we obtain the truth of the Riemann-Hardy conjecture.
Keywords
Ramanujan zeta-function; Ramanujan's tau-function; entire Ramanujan zeta-function; Ramanujan $\Xi$-function; Riemann-Hardy conjecture
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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