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Unnormalized differences between zeros of L-functions

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  24 October 2014

Kevin Ford  and
Alexandru Zaharescu
Kevin Ford
Affiliation:
Department of Mathematics, 1409 West Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email ford@math.uiuc.edu
Alexandru Zaharescu
Affiliation:
Department of Mathematics, 1409 West Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email zaharesc@math.uiuc.edu
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Abstract

We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function, which was observed by a number of authors. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general $L$-functions. In particular, we show how the rank of an elliptic curve over $\mathbb{Q}$ is encoded in the sequences of zeros of other$L$-functions, not only the one associated to the curve.

Keywords

Riemann zeta functionL-functionsdistribution of zeros

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11K38: Irregularities of distribution, discrepancy

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 2 , February 2015 , pp. 230 - 252
Copyright
© The Author(s) 2014 

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