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Unnormalized differences between zeros of L-functions

Published online by Cambridge University Press:  24 October 2014

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

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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.

Type
Research Article
Copyright
© The Author(s) 2014 

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