A note on series equivalent of the Riemann hypothesis

Shekhar Suman and Raman Kumar Das ()
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Shekhar Suman: Ranchi University
Raman Kumar Das: St. Xavier’s College

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 1, 117-119

Abstract: Abstract In this manuscript we denote by $$\sum _{\rho }$$ ∑ ρ a sum over the non trivial zeros of Riemann zeta function (or over the zeros of Riemann’s xi function), where the zeros of multiplicity k are counted k times. We prove a result that the Riemann Hypothesis is true if and only if $$\begin{aligned} \sum _{\rho }\frac{1}{|\frac{1}{2}-\rho |^2}=\frac{\xi ''(\frac{1}{2})}{\xi (\frac{1}{2})} \end{aligned}$$ ∑ ρ 1 | 1 2 - ρ | 2 = ξ ′ ′ ( 1 2 ) ξ ( 1 2 )

Keywords: Riemann zeta function; Riemann xi function; Riemann Hypothesis; Hadamard product; 11M26; 11M06; 11M32 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00237-6

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