Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-14T18:35:55.376Z Has data issue: false hasContentIssue false
  • English
  • Français

A Discrete Mean Value of the Derivative of the Riemann Zeta Function

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

Published online by Cambridge University Press:  21 December 2009

Nathan Ng
Nathan Ng
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, ON, Canada, K1N 6N5.
Get access

Abstract

In this article, a discrete mean value of the derivative of the Riemann zeta function is computed. This mean value will be important for several applications concerning the size of ζ′(ρ), where ζ(s) is the Riemann zeta function and ρ is a non-trivial zero of ζ(s).

MSC classification

Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 113 - 155
Copyright
Copyright © University College London 2007

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

1Conrey, J. B., Ghosh, A. and Gonek, S. M., Large gaps between zeros of the zeta-function. Mathematika 33 (1986), 212–238.CrossRefGoogle Scholar
2Conrey, J. B., Ghosh, A. and Gonek, S. M., Simple zeros of zeta functions. In Colloque de Théorie Analytique des Nombres “Jean Coquet” (Marseille, 1985), Publ. Math. Orsay, 88–02, Univ. Paris XI (Orsay, 1988), 77–83.Google Scholar
3Conrey, J. B., Ghosh, A. and Gonek, S. M., Simple zeros of the Riemann zeta function. Proc. London Math. Soc. (3) 76 (1998), 497–522.CrossRefGoogle Scholar
4Davenport, H., Multiplicative number theory (3rd ed, revised by H. L. Montgomery). Graduate Texts in Mathematics 74, Springer (New York, 1980).CrossRefGoogle Scholar
5Fujii, A., On a conjecture of Shanks. Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 109–114.CrossRefGoogle Scholar
6Gallagher, P. X., Bombieri's mean value theorem. Mathematika 15 (1968), 1–6.CrossRefGoogle Scholar
7Gonek, S. M., Mean values of the Riemann zeta function and its derivatives. Invent. Math. 75 (1984), 123–141.CrossRefGoogle Scholar
8Montgomery, H. L., Topics in Multiplicative Number Theory. Lecture Notes in Mathematics 227, Springer (Berlin, 1971).CrossRefGoogle Scholar
9Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory, 1: Classical Theory. Cambridge University Press (Cambridge, 2006).CrossRefGoogle Scholar
10Ng, N., The fourth moment of ζ′(ρ). Duke Math. J. 125 (2004), 243–266.CrossRefGoogle Scholar
11Ng, N., The distribution of the summatory function of the Möbius function. Proc. London Math. Soc. (3) 89 (2004), 361–369.CrossRefGoogle Scholar
12Ng, N., Large gaps between the zeros of the Riemann zeta function. J. Number Theory. 128 (2008), 509–556.CrossRefGoogle Scholar
13Ng, N., Extreme values of ζ′(ρ) (available at http://arxiv.org/abs/0706.1765), J. London Math. Soc. (to appear).Google Scholar
14Rudnick, Z. and Soundararajan, K., Lower bounds for moments of L-functions. Proc. Nat. Acad. Sci. U.S.A. 102 (2005), 6837–6838.CrossRefGoogle ScholarPubMed
15Rudnick, Z. and Soundararajan, K., Lower bounds for moments of L-functions: symplectic and orthogonal examples. In Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory (Ed. Friedberg, , Bump, , Goldfeld, , and Hoffstein, ), Proc. Symp. Pure Math., 75, Amer. Math. Soc. (2006).Google Scholar
16Shiu, P., A Brun-Titchmarsh theorem for multiplicative functions. J. reine angew. Math. 313 (1980), 161–170.Google Scholar
17Soundararajan, K., Extreme values of L-functions at the central point (available at http://arxiv.org/abs/0708.3990), Math. Annalen (to appear).Google Scholar
18Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press (Cambridge, 1995).Google Scholar
19Vaughan, R. C., Mean value theorems in prime number theory. J. London Math. Soc. (2) 10 (1975), 153–162.CrossRefGoogle Scholar