Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T16:41:55.796Z Has data issue: false hasContentIssue false
  • English
  • Français

NEGATIVE VALUES OF THE RIEMANN ZETA FUNCTION ON THE CRITICAL LINE

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

Published online by Cambridge University Press:  17 June 2013

Justas Kalpokas ,
Maxim A. Korolev  and
Jörn Steuding
Justas Kalpokas
Affiliation:
Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius,Lithuania email [email protected]
Maxim A. Korolev
Affiliation:
Steklov Mathematical Institute, Gubkina str. 8, 119991 Moscow,Russia email [email protected], [email protected]
Jörn Steuding
Affiliation:
Department of Mathematics, Würzburg University, Am Hubland, 97218 Würzburg,Germany email [email protected]
Get access

Abstract

We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

MSC classification

Secondary: 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 443 - 462
Copyright
Copyright © University College London 2013 

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.)

References

Christ, T. and Kalpokas, J., Upper bounds for discrete moments of the derivatives of the Riemann zeta-function on the critical line. Lith. Math. J. 52 (3) (2012), 233–248.CrossRefGoogle Scholar
Edwards, H. M., Riemann’s Zeta Function, Academic Press (New York, 1974).Google Scholar
Heath-Brown, D. R., Fractional moments of the Riemann zeta-function. J. Lond. Math. Soc. (2) 24 (1981), 65–78.CrossRefGoogle Scholar
Gonek, S. M., Mean values of the Riemann zeta-function and its derivatives. Invent. Math. 75 (1) (1984), 123–141.CrossRefGoogle Scholar
Gram, J., Sur les zéros de la fonction $\zeta (s)$ de Riemann. Acta Math. 27 (1903), 289–304.CrossRefGoogle Scholar
Israilov, M. I., On the Laurent expansion of the Riemann zeta-function. Proc. Steklov Inst. Math. 158 (1983), 105–112.Google Scholar
Ivić, A., The Riemann Zeta-Function, John Wiley & Sons (New York, 1985).Google Scholar
Kalpokas, J. and Steuding, J., On the value-distribution of the Riemann zeta-function on the critical line. Mosc. J. Combin. Number Theory 1 (2011), 26–42.Google Scholar
Milinovich, M. B. and Ng, N., Lower bounds for moments of ${\zeta }^{\prime } (\rho )$, Int. Math. Res. Not. (2013), doi:10.1093/imrn/rnt028.CrossRefGoogle Scholar
Ng, N., A discrete mean value of the derivative of the Riemann zeta function. Mathematika 54 (2007), 113–155.CrossRefGoogle Scholar
Rudnick, Z. and Soundararajan, K., Lower bounds for moments of L-functions. Proc. Natl. Sci. Acad. USA 102 (2005), 6837–6838.CrossRefGoogle ScholarPubMed
Soundararajan, K., Extreme values of zeta and L-functions. Math. Ann. 342 (2008), 467–486.CrossRefGoogle Scholar
Titchmarsh, E. C., On van der Corput’s method and the zeta-function of Riemann, IV. Quart. J. Math. 5 (1934), 98–105.CrossRefGoogle Scholar