Abstract
We prove that more than nine percent of the central values $L(1/2,\\chi_p)$ are non-zero, where $p\\equiv 1 \\pmod{8}$ ranges over primes and $\\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(1/2,\\chi_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.
Highlights
Introduction and resultsThe values of L-functions at special points on the complex plane are of great interest.At the fixed point of the functional equation, called the central point, the question of nonvanishing is important
The well-known Birch and Swinnerton-Dyer conjecture [43] relates the order of vanishing of certain L-functions at the central point to the arithmetic of elliptic curves
The analysis of each family they discuss leads to a Density Conjecture that, if true, would imply that almost all L-functions in the family do not vanish at the central point
Summary
The values of L-functions at special points on the complex plane are of great interest. Balasubramanian and Murty [3] were the first to prove that a (small) positive proportion of this family does not vanish at the central point They used the celebrated technique of mollified moments, a method that has been highly useful in other contexts (see, for example, [4, 9, 38]). Jutila [21] initiated the study of non-vanishing at the central point for this family and proved that for infinitely many fundamental discriminants d. Our work indicates that Soundararajan’s lower bound [39] for the proportion of nonvanishing for fundamental discrimimants d ≡ 0 (mod 8) holds for the case of fundamental discriminants d ≡ 1 (mod 8) Proving this involves re-doing the calculations, but without applying an upper bound sieve.
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