Partial Euler Products on the Critical Line

Conrad, Keith

Canadian journal of mathematics, Tome 57 (2005) no. 2, pp. 267-297

Résumé

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.

Détail

DOI : 10.4153/CJM-2005-012-6

Mots-clés : 11M41, 11S40, Euler product, explicit formula, second moment

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