Hostname: page-component-f554764f5-68cz6 Total loading time: 0 Render date: 2025-04-20T14:04:22.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on a formula of Ramanujan

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  06 February 2024

Andrés Chirre Open the ORCID record for Andrés Chirre [Opens in a new window]  and
Steven M. Gonek Open the ORCID record for Steven M. Gonek [Opens in a new window]
Andrés Chirre
Affiliation:
Departamento de Ciencias - Sección Matemáticas, Pontificia Universidad Católica del Perú, Lima, Peru ([email protected])
Steven M. Gonek
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.

Keywords

Riemann zeta functionnontrivial zerosRiemann hypothesisMöbius μ function

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 12
Check for updates
Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Agarwal, A., Garg, M. and Maji, B.. Riesz-type criteria for the Riemann hypothesis. Proc. Am. Math. Soc. 150 (2022), 5151–5163.Google Scholar
Dixit, A.. Character analogues of Ramanujan-type integrals involving the Riemann $\Xi$-function. Pac. J. Math. 255 (2012), 317–348.Google Scholar
Dixit, A.. Analogues of the general theta transformation formula. Proc. R. Soc. Edinburgh 143A (2013), 371–399.CrossRefGoogle Scholar
Hardy, G. H. and Littlewood, J. E.. Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Math. 41 (1916), 119–196.Google Scholar
Juyal, A., Maji, B. and Sathyanarayana, S.. An exact formula for a Lambert series associated to a cusp form and the Möbius function. Ramanujan J. 57 (2022), 769–784.CrossRefGoogle Scholar
Kühn, P., Robles, N. and Roy, A.. On a class of functions that satisfies explicit formulae involving the Möbius function. Ramanujan J. 38 (2015), 383–422.CrossRefGoogle Scholar
Paris, R. B.. The numerical evaluation of the Riesz function. arXiv:2107.02800 [math.CA].Google Scholar
Riesz, M.. Sur l'hypothèse de Riemann. Acta Math. 40 (1916), 185–190.CrossRefGoogle Scholar
Roy, A., Zaharescu, A. and Zaki, M.. Some identities involving convolutions of Dirichlet characters and the Möbius function. Proc. Indian Acad. Sci. Math. Sci. 126 (2016), 21–33.Google Scholar
Staś, W.. Zur theorie der Möbiusschen $\mu$-funktion (German). Acta Arith. 7 (1962), 409–416.Google Scholar
Staś, W.. Über eine Reihe von Ramanujan (German). Acta Arith. 8 (1963), 261–271.Google Scholar
Staś, W.. Some remarks on a series of Ramanujan. Acta Arith. 10 (1965), 359–368.CrossRefGoogle Scholar
Titchmarsh, E. C.. The theory of the Riemann zeta-function, 2nd edn (The Clarendon Press, Oxford University Press, New York, 1986).Google Scholar