Submitted:
29 May 2021
Posted:
31 May 2021
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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
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