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

TL;DR

Under certain hypotheses, the paper proves that sums of coefficients of general L-functions in short intervals follow a normal distribution, extending previous results to the degree aspect.

Contribution

It introduces a new central limit theorem for L-function coefficients in short intervals, especially in the degree aspect under GLH, generalizing prior lattice point results.

Findings

Establishes normality of L-function coefficients sums in short intervals.

Extends Hughes and Rudnick's lattice point results to a broader L-function context.

Operates under assumptions like GLH, weak Ramanujan, and partial sum estimates.

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.