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Zeroes of zeta functions and symmetry

Authors: Nicholas M. Katz and Peter Sarnak
Journal: Bull. Amer. Math. Soc. 36 (1999), 1-26
MSC (1991): Primary 11G, 11M, 11R, 11Y; Secondary 60B, 81Q
MathSciNet review: 1640151
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Abstract: Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the ``function field'' analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and $L$-functions.

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Additional Information

Nicholas M. Katz
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544

Peter Sarnak
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544

Received by editor(s): October 15, 1997
Received by editor(s) in revised form: August 28, 1998
Additional Notes: Research partially supported by NSF grants DMS 9506412 and DMS 9401571.
Article copyright: © Copyright 1999 American Mathematical Society