Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T03:33:27.293Z Has data issue: false hasContentIssue false
  • English
  • Français

On Negative Moments of the Riemann Zeta-Function

Published online by Cambridge University Press:  26 February 2010

S. M. Gonek
S. M. Gonek
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A.
Get access

Extract

The purpose of this paper is to take some first steps the investigation of the negative moments

where k>0 and 12, and the related discrete moments

where runs over the complex zeros of the zeta-function. We assume the Riemann hypothesis (RH) throughout; it then follows that Ik(, T) converges for every k > 0 when > but for no k = when =. We further note that Jk(T) is only defined for all T if all the zeros are simple and, in that case, Ik(, T) converges for all k<.

Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 71 - 88
Copyright
Copyright University College London 1989

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

1.Conrey, J. B. and Ghosh, A.. On mean values of the zeta-function. Mathematika, 31 (1984), 159161.Google Scholar
2.Gonek, S. M.. Mean values of the Riemann zeta-function and its derivatives. Invent. Math., 74 (1984), 123141.Google Scholar
3.Heath-Brown, D. R.. Fractional moments of the Riemann zeta-function. J. London Math. Soc., (2), 24 (1981), 6578.Google Scholar
4.Hejhal, D. A.. On the distribution of log Number Theory, Trace Formulas and Discrete Groups, Symposium in Honor ofAtle Selberg, Oslo, Norway, July 14-21, 1987, edited by Aubert, , Bombieri and , Goldfeld (Academic Press, San Diego, 1989), 343370.Google Scholar
5.Titchmarsh, E. C.. The Theory of the Riemann Zeta-function second edition revised by Heath-Brown, D. R. (Clarendon Press, Oxford, 1986).Google Scholar