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Search: MSC category 11S40 ( Zeta functions and $L$-functions [See also 11M41, 19F27] )

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1. CJM 2005 (vol 57 pp. 267)

 Partial Euler Products on the Critical Line 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. Keywords:Euler product, explicit formula, second momentCategories:11M41, 11S40