Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-05-06T08:02:18.106Z Has data issue: false hasContentIssue false
  • English
  • Français

SUM OF VALUES OF THE IDEAL CLASS ZETA-FUNCTION OVER NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

Part of: Algebraic number theory: global fields Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  31 July 2023

SAEREE WANANIYAKUL Open the ORCID record for SAEREE WANANIYAKUL [Opens in a new window] ,
JÖRN STEUDING  and
NITHI RUNGTANAPIROM Open the ORCID record for NITHI RUNGTANAPIROM [Opens in a new window]
SAEREE WANANIYAKUL
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand e-mail: [email protected]
JÖRN STEUDING
Affiliation:
Department of Mathematics, Würzburg University, Am Hubland, Würzburg 97218, Germany e-mail: [email protected]
NITHI RUNGTANAPIROM*
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove an upper bound for the sum of values of the ideal class zeta-function over nontrivial zeros of the Riemann zeta-function. The same result for the Dedekind zeta-function is also obtained. This may shed light on some unproved cases of the general Dedekind conjecture.

Keywords

Dedekind’s conjecturenontrivial zeros of Riemann zeta-functionzeta-function

MSC classification

Primary: 11R42: Zeta functions and $L$-functions of number fields
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 288 - 300
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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

Apostol, T. M., Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics (Springer, New York, 1976).Google Scholar
Brauer, R., ‘On the zeta-functions on algebraic number fields’, Amer. J. Math. 69(2) (1947), 243–250.CrossRefGoogle Scholar
Conrey, J. B., Ghosh, A. and Gonek, S. M., ‘Simple zeros of the zeta function of a quadratic number field. I’, Invent. Math. 86 (1986), 563–576.CrossRefGoogle Scholar
Conrey, J. B., Ghosh, A. and Gonek, S. M., ‘Simple zeros of the zeta function of a quadratic number field. II’, in: Analytic Number Theory and Diophantine Problems, Proc. Conf., Stillwater, Oklahoma, 1984, Progress in Mathematics, 70 (eds. Adolphson, A. C., Conrey, J. B., Ghosh, A. and Yager, R. I.) (Birkhäuser, Boston, MA, 1987), 87–114.Google Scholar
Conrey, J. B., Ghosh, A. and Gonek, S. M., ‘Simple zeros of zeta functions’, Publ. Math. Orsay 88 (1988), 77–83.Google Scholar
Garunkštis, R. and Kalpokas, J., ‘Sum of the periodic zeta-function over the nontrivial zeros of the Riemann zeta-function’, Analysis 28(2) (2008), 209–217.CrossRefGoogle Scholar
Gonek, S. M., ‘Mean values of the Riemann zeta-function and its derivatives’, Invent. Math. 75 (1984), 123–141.CrossRefGoogle Scholar
Lang, S., Algebraic Number Theory, 2nd edn, Graduate Texts in Mathematics, 110 (Springer, New York, 1994).CrossRefGoogle Scholar
Murty, M. R. and Murty, V. K., Non-Vanishing of $L$ -Functions and Applications, Progress in Mathematics, 157 (Birkhäuser, Basel, 1997).Google Scholar
Paul, B. and Sankaranarayanan, A., ‘On the error term and zeros of the Dedekind zeta function’, J. Number Theory 215 (2020), 98–119.CrossRefGoogle Scholar
Steuding, J., ‘On the value distribution of Hurwitz zeta-functions at the nontrivial zeros of the Riemann zeta-function’, Abh. Math. Semin. Univ. Hambg. 71 (2001), 113–121.CrossRefGoogle Scholar
Titchmarsh, E. C., The Theory of Functions, 2nd edn (Oxford University Press, Oxford, 1939).Google Scholar
Titchmarsh, E. C., The Theory of the Riemann Zeta-function, 2nd edn (Oxford University Press, Oxford, 1986).Google Scholar
Tongsomporn, J., Wananiyakul, S. and Steuding, J., ‘The values of the periodic zeta-function at the nontrivial zeros of Riemann’s zeta-function’, Symmetry 13(12) (2021), 2410.CrossRefGoogle Scholar
Uchida, K., ‘On Artin $L$ -functions’, Tohoku Math. J. (2) 27 (1975), 75–81.CrossRefGoogle Scholar
van der Waall, R. W., ‘On a conjecture of Dedekind on zeta-functions’, Indag. Math. (N.S.) 78 (1975), 83–86.CrossRefGoogle Scholar