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Partial Euler Products on the Critical Line

Published online by Cambridge University Press:  20 November 2018

Keith Conrad*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, U.S.A. e-mail: kconrad@math.uconn.edu
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Abstract

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The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve $L$-function at $s\,=\,1$. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the $L$-function and that the constant in the asymptotics has an unexpected factor of $\sqrt{2}$. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical $L$-function along its critical line. The general $\sqrt{2}$ phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seems much deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2005

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