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AN ARITHMETIC EQUIVALENCE OF THE RIEMANN HYPOTHESIS

Part of: Multiplicative number theory Number theory Elementary number theory

Published online by Cambridge University Press:  18 June 2018

MARC DELÉGLISE  and
JEAN-LOUIS NICOLAS
MARC DELÉGLISE*
Affiliation:
Univ Lyon, Université Claude Bernard, Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Mathématiques, Bât. Doyen Jean Braconnier, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne cedex, France email m.h.deleglise@gmail.com
JEAN-LOUIS NICOLAS
Affiliation:
Univ Lyon, Université Claude Bernard, Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Mathématiques, Bât. Doyen Jean Braconnier, 43 Bd du 11 Novembre 1918, F-69622 Villeurbanne cedex, France email nicolas@math.univ-lyon1.fr
*
✉m.h.deleglise@gmail.com
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Abstract

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Let $h(n)$ denote the largest product of distinct primes whose sum does not exceed $n$. The main result of this paper is that the property for all $n\geq 1$, we have $\log h(n)<\sqrt{\text{li}^{-1}(n)}$ (where $\text{li}^{-1}$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.

Keywords

arithmetic functionRiemann hypothesisLandau function

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11N37: Asymptotic results on arithmetic functions 11N05: Distribution of primes 11-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 2 , April 2019 , pp. 235 - 273
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

Research partially supported by CNRS, Institut Camille Jordan, UMR 5208.

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