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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 justas.kalpokas@mif.vu.lt
Maxim A. Korolev
Affiliation:
Steklov Mathematical Institute, Gubkina str. 8, 119991 Moscow,Russia email hardy_ramanujan@mail.ru, korolevma@mi.ras.ru
Jörn Steuding
Affiliation:
Department of Mathematics, Würzburg University, Am Hubland, 97218 Würzburg,Germany email steuding@mathematik.uni-wuerzburg.de
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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 

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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