Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T19:50:24.298Z Has data issue: false hasContentIssue false
  • English
  • Français

A simpler proof of Levinson's theorem

Published online by Cambridge University Press:  24 October 2008

J. B. Conrey  and
A. Ghosh
J. B. Conrey
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A.
A. Ghosh
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], let

where Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and let

where

and

where

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 385 - 395
Copyright
Copyright © Cambridge Philosophical Society 1985

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

REFERENCES

[1]Conrey, J. B., Ghosh, A. and Gonek, S. M.. Simple zeros of the Riemann zeta-function (Submitted for publication).Google Scholar
[2]Estermann, T.. On the representation of a number as the sum of two products. Proc. London Math. Soc. (2) 31 (1930), 123–133.CrossRefGoogle Scholar
[3]Levinson, N.. More than one third of zeros of Riemann's zeta-function are on σ = ½. Adv. in Math. 13 (1974), 383–436.CrossRefGoogle Scholar
[4]Vaughan, R. C.. Mean value theorems in prime number theory. J. London Math. Soc. (2) 10 (1975), 153–162.CrossRefGoogle Scholar