Weighted central limit theorems for central values of $L$-functions

Abstract

We establish a central limit theorem for the central values of Dirichlet Lfunctions with respect to a weighted measure on the set of primitive characters modulo q as q → ∞. Under the Generalized Riemann Hypothesis (GRH), we also prove a weighted central limit theorem for the joint distribution of the central L-values corresponding to twists of two distinct primitive Hecke eigenforms. As applications, we obtain (under GRH) positive proportions of twists for which the central L-values simultaneously grow or shrink with q as well as a positive proportion of twists for which linear combinations of the central L-values are nonzero.

Conditional lower bounds on the distribution of central values in families of $L$-functions

“…Here we interpret log L 1 2 , E d to be negative infinity if L 1 2 , E d = 0, and the conjecture implies in particular that L 1 2 , E d ̸ = 0 for almost all d ∈ E. Towards this conjecture, we established in [7] that N (X; α, ∞) is bounded above by the right hand side of the conjectured relation (1.2). Complementing this, we now establish a conditional lower bound for N (X; α, β).…”

“…Introduction. Selberg [11,12] (see [8] for a recent treatment) established that if t is chosen uniformly from [0, T ] then the values log ζ 1 2 + it are distributed approximately like a Gaussian random variable with mean 0 and variance 1 2 log log T . More recently, Keating and Snaith [6] have conjectured that central values in families of L-functions have an analogous log-normal distribution with a prescribed mean and variance depending on the "symmetry type" of the family.…”

“…The Keating-Snaith conjectures predict that for d ∈ E, the quantity log L 1 2 , E d has an approximately normal distribution with mean − 1 2 log log |d| and variance log log |d|. To state this precisely, let α < β be real numbers, and for any X ≥ 20, define…”

“…We end the introduction by mentioning the recent work of Bui, Evans, Lester, and Pratt [1] who establish "weighted" (where the weight is a mollified central value) analogues of the Keating-Snaith conjecture. This amounts to a form of conditioning on non-zero value since central values that are zero are assigned a weight equal to zero.…”

“…This amounts to a form of conditioning on non-zero value since central values that are zero are assigned a weight equal to zero. The use of such a weighted measure allows [1] to establish a full asymptotic, however as a side effect they have little control over the nature of the weight.…”