A central limit theorem for coefficients of $L$-functions in short intervals

Abstract

Assuming the generalized Lindel\"{o}f hypothesis (GLH), a weak version of the generalized Ramanujan conjecture and a Rankin--Selberg type partial sum estimate, we establish the normality of the sum of coefficients of a general $L$-function in short intervals of appropriate length. The novelty lies in the degree aspect under GLH. In particular, this generalizes the result of Hughes and Rudnick on lattice point counts in thin annuli.