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Partial Euler Products on the Critical Line
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 2 / 01 April 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 267-297
- Print publication:
- 01 April 2005
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- Article
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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.
