A sequential Riesz-like criterion for the Riemann hypothesis

Luis Báez-Duarte

International Journal of Mathematics and Mathematical Sciences, 2005, vol. 2005, 1-11

Abstract:

Let c k : = ∑ j − 0 k ( − 1 ) j ( k j ) ( 1 / ζ ( 2 j + 2 ) ) . We prove that the Riemann hypothesis is equivalent to c k ≪ k − 3 / 4 + ε for all ε > 0 ; furthermore, we prove that c k ≪ k − 3 / 4 implies that the zeros of ζ ( s ) are simple. This is closely related to M. Riesz's criterion which states that the Riemann hypothesis is equivalent to ∑ k = 1 ∞ ( ( − 1 ) k + 1 x k / ( k − 1 ) ! ζ ( 2 k ) ) ≪ x 1 / 4 + ε as x → + ∞ , for all ε > 0 .

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:141565

DOI: 10.1155/IJMMS.2005.3527

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