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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.
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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

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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