HYPOTHESIS | doi:10.20944/preprints202105.0769.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Ramanujan zeta-function; Ramanujan's tau-function; entire Ramanujan zeta-function; Ramanujan $\Xi$-function; Riemann-Hardy conjecture
Online: 31 May 2021 (12:34:12 CEST)
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.
ARTICLE | doi:10.20944/preprints202307.0252.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Ramanujan's theta function; Modular equation; $q$-Shifted factorial
Online: 5 July 2023 (03:39:26 CEST)
We establish certain modular equations of degree three based on some results of Ramanujan's theta functions.