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Simple zeros of the Riemann zeta-function

Published online by Cambridge University Press:  01 May 1998

JB Conrey ,
A Ghosh  and
SM Gonek
JB Conrey
Affiliation:
Department of Mathematics, Oklahoma State University, College of Arts & Science, 401 Mathematical Sciences, Stillwater, OK 74078-0613, USA. E-mail: conrey@math.okstate.edu, ghosh@math.okstate.edu
A Ghosh
Affiliation:
Department of Mathematics, Oklahoma State University, College of Arts & Science, 401 Mathematical Sciences, Stillwater, OK 74078-0613, USA. E-mail: conrey@math.okstate.edu, ghosh@math.okstate.edu
SM Gonek
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, USA. E-mail: gonek@math.rochester.edu
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Abstract

Assuming the Riemann Hypothesis, Montgomery showed by means of his pair correlation method that at least two-thirds of the zeros of Riemann's zeta-function are simple. Later he and Taylor improved this to 67.25 percent and, more recently, Cheer and Goldston increased the percentage to 67.2753. Here we prove by a new method that if the Riemann and Generalized Lindel\"of Hypotheses hold, then at least 70.3704 percent of the zeros are simple and at least 84.5679 percent are distinct. Our method uses mean value estimates for various functions defined by Dirichlet series sampled at the zeros of the Riemann zeta-function.

1991 Mathematics Subject Classification: 11M26.

Keywords

Riemann's zeta-functionsimple zerosmean value theorems
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 76 , Issue 3 , May 1998 , pp. 497 - 522
Copyright
London Mathematical Society 1998

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