Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T17:39:30.339Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ARITHMETIC EQUIVALENCE OF THE RIEMANN HYPOTHESIS

Part of: Number theory Multiplicative 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions (Dover Publications, New York).Google Scholar
Axler, C., ‘New bounds for the sum of the first $n$ prime numbers’, J. Théor. Nombres Bordeaux (to appear) (2016), arXiv:1606.06874.Google Scholar
Axler, C., ‘New estimates for some functions defined over primes’, Intergers (to appear) (2017),arXiv:1703.08032.Google Scholar
Berndt, B. C., Ramanujan’s Notebooks, Part IV (Springer, New York, 1994), 126–131.Google Scholar
Cohen, H., Number Theory, Analytic and Modern Tools, Volume II (Springer, New York, 2007).Google Scholar
Deléglise, M. and Nicolas, J.-L., ‘Le plus grand facteur premier de la fonction de Landau’, Ramanujan J. 27 (2012), 109–145.Google Scholar
Deléglise, M. and Nicolas, J.-L., ‘Maximal product of primes whose sum is bounded’, Proc. Steklov Inst. Math. 282 (2013), 73–102; (Issue 1, Supplement).Google Scholar
Deléglise, M. and Nicolas, J.-L., ‘On the largest product of primes with bounded sums’, J. Integer Sequences 18 (2015), Article 15.2.8.Google Scholar
Deléglise, M., Nicolas, J.-L. and Zimmermann, P., ‘Landau’s function for one million billions’, J. Théor. Nombres Bordeaux 20 (2008), 625–671.Google Scholar
Deléglise, M. and Rivat, J., ‘Computing 𝜋(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method’, Math. Comp. 65 (1996), 235–245.Google Scholar
Deléglise, M. and Rivat, J., ‘Computing 𝜓(x)’, Math. Comp. 67 (1998), 1691–1696.Google Scholar
Dusart, P., ‘Explicit estimates of some functions over primes’, Ramanujan J. 45 (2018), 227–251.Google Scholar
Edwards, H. M., Riemann’s Zeta Function (Academic Press, New York, 1974).Google Scholar
Ellison, W. J. and Ellison, F., Prime Numbers (Wiley, New York, 1985).Google Scholar
Ellison, W. J. and Mendès France, M., Les nombres premiers (Hermann, Paris, 1975).Google Scholar
Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen, I, 2nd edn (Chelsea, New York, 1953).Google Scholar
Massias, J.-P. and Robin, G., ‘Bornes effectives pour certaines fonctions concernant les nombres premiers’, J. Théor. Nombres Bordeaux 8 (1996), 215–242.Google Scholar
Massias, J.-P., Nicolas, J.-L. and Robin, G., ‘Évaluation asymptotique de l’ordre maximum d’un élément du groupe symétrique’, Acta Arith. 50 (1988), 221–242.Google Scholar
Massias, J.-P., Nicolas, J.-L. and Robin, G., ‘Effective bounds for the maximal order of an element in the symmetric group’, Math. Comp. 53(188) (1989), 665–678.Google Scholar
Nicolas, J.-L., ‘Small values of the Euler function and the Riemann hypothesis’, Acta Arith. 155 (2012), 311–321.Google Scholar
Platt, D. J. and Trudgian, T., ‘On the first sign change of 𝜃(x) - x ’, Math. Comp. 85(299) (2016), 1539–1547.Google Scholar
Robin, G., ‘Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann’, J. Math. Pures Appl. 63 (1984), 187–213.Google Scholar
Salvy, B., ‘Fast computation of some asymptotic functional inverses’, J. Symbolic Comput. 17 (1994), 227–236.Google Scholar
Schoenfeld, L., ‘Sharper bounds for the Chebyshev functions 𝜃(x) and 𝜓(x) II’, Math. Comp. 30 (1976), 337–360.Google Scholar
https://github.com/mhdeleglise/PsiTheta.Google Scholar
http://math.univ-lyon1.fr/homes-www/deleglis/calculs.html.Google Scholar
http://math.univ-lyon1.fr/homes-www/nicolas/calculhdenHR.html.Google Scholar
http://www.dtc.umn.edu/ odlyzko/zeta_tables/zeros2.Google Scholar