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EULER PRODUCT ASYMPTOTICS FOR DIRICHLET $\boldsymbol {L}$-FUNCTIONS

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

Published online by Cambridge University Press:  07 January 2022

IKUYA KANEKO Open the ORCID record for IKUYA KANEKO [Opens in a new window]
IKUYA KANEKO*
Affiliation:
Department of Mathematics, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
*
e-mail: [email protected]
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Abstract

The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.

Keywords

Dirichlet L-functionspartial Euler productsdeep Riemann hypothesis

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
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 48 - 56
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Akatsuka, H., ‘The Euler product for the Riemann zeta-function in the critical strip’, Kodai Math. J. 40(1) (2017), 79–101.CrossRefGoogle Scholar
Conrad, K., ‘Partial Euler products on the critical line’, Canad. J. Math. 57(2) (2005), 267–297.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Kaneko, I. and Koyama, S., ‘Euler products of Selberg zeta functions in the critical strip’, Preprint, 2018, arXiv:1809.10140.Google Scholar
Kuo, W. and Murty, M. R., ‘On a conjecture of Birch and Swinnerton-Dyer’, Canad. J. Math. 57(2) (2005), 328–337.Google Scholar
Mertens, F., ‘Ein Beitrag zur analytischen Zahlentheorie’, J. reine angew. Math. 78 (1874), 46–62.Google Scholar
Nicolas, J.-L., ‘The sum of divisors function and the Riemann hypothesis’, Ramanujan J. (2021), doi:10.1007/s11139-021-00491-y.Google Scholar
Ramanujan, S., ‘Highly composite numbers’, Proc. Lond. Math. Soc. (2) 14 (1915), 347–409.CrossRefGoogle Scholar
Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa Publishing House, New Delhi, 1988).Google Scholar
Ramanujan, S., ‘Highly composite numbers’, Ramanujan J. 1(2) (1997), 119–153, annotated and with a foreword by Jean-Louis Nicolas and Guy Robin.Google Scholar
Siegel, C. L., ‘Über die Classenzahl quadratischer Zahlkörper’, Acta Arith. 1 (1935), 83–86.Google Scholar
Walfisz, A., ‘Zur additiven Zahlentheorie. II’, Math. Z. 40 (1936), 92–607.CrossRefGoogle Scholar