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 moment Euler product, explicit formula, second moment

MSC Classifications:

11M41, 11S40 show english descriptions Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Zeta functions and $L$-functions [See also 11M41, 19F27] 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
11S40 - Zeta functions and $L$-functions [See also 11M41, 19F27]

top of page | contact us | social media | privacy | site map