Partial Euler Products on the Critical Line

Abstract

Abstract. 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 √ 2. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general √ 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.

Ranks of quadratic twists of elliptic curves

“…Also, noting that the maximum ofF (t) −B 10 (t) is about 0.0226 near t ≈ 8.45, whileB 10 (t) exceedsF (t) by twice this amount around t ≈ 4, the placing of zeros must be rather delicate to simultaneously have both γF (Sγ) ≈ 7.1379 and γB 10 (Sγ) ≈ 7.0901. Looking at Figure 2, one sees that the minor contributions from distant zeros are approximately halved withB 10 . For instance, there should be approximately 4 zeros between 4π and 6π, contributing maybe 0.02 more toF thanB 10 , this then being doubled by evenness.…”

“…9 The lattice displayed here is also given by Rubin and Silverberg in [50, §9]. 10 Indeed, trivially (d1, d2, d3, d4) = (1, 1, 1, 1) generates all (u, v), but the "short" vector enumeration for lattices of such small determinant is typically infeasible. The exact correspondence between this method and that given in the previous subsection has not been fleshed out completely.…”

“…The work of Nagao [41, §3] contains the correct balance with 1/2. See [10] for the general context. 14 This model for the ap is probably most useful when Y is sufficiently smaller than √ N , else effects from modularity of the L-function may play a part.…”

On a Conjecture of Birch and Swinnerton-Dyer

“…His proof uses the explicit formula method. Our proof invokes a beautiful, little known theorem of Marcel Riesz [9] and is shorter than the one in [4]. Our approach can also be used to treat the general context.…”

“…After this work was done, K. Murty informed us that Keith Conrad [4] has also proved Theorem 3 in the more general context of L-functions with Euler products of GL(n)-type. His proof uses the explicit formula method.…”