Central limit theorems for sums of quadratic characters, Hecke eigenforms, and elliptic curves

Murty, M. Ram; Prabhu, Neha

doi:10.1090/proc/14760

Central limit theorems for sums of quadratic characters, Hecke eigenforms, and elliptic curves
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by M. Ram Murty and Neha Prabhu PDF

Proc. Amer. Math. Soc. 148 (2020), 965-977 Request permission

Abstract:

We prove central limit theorems (under suitable growth conditions) for sums of quadratic characters, families of Hecke eigenforms of level $1$ and weight $k$, and families of elliptic curves, twisted by an $L$-function satisfying certain properties. As a corollary, we obtain a central limit theorem for products $\chi (p)a_f(p)$ where $\chi$ is a quadratic Dirichlet character and $f$ is a normalized Hecke eigenform.

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