Large deviations of sums of random variables

TL;DR

This paper studies the probabilities of large deviations in sums of weighted, approximately independent random variables, with applications to number theory sequences like primes, characters, cusp forms, and Kloosterman sums.

Contribution

It generalizes and improves existing large deviation results for sums of weighted, approximately independent variables, motivated by diverse number theory examples.

Findings

Provides new bounds for large deviations in number-theoretic sequences.

Extends previous results to more general weighted sums.

Applications include prime numbers, characters, cusp forms, and Kloosterman sums.

Abstract

In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some of the results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences $p^{it}$, $\chi(p)$, $\chi_d(p)$, $\lambda_f(p)$, and $\text{Kl}_q(a-n, b)$; where $p$ ranges over the primes, $t$ varies in a large interval, $\chi$ varies among all characters modulo $q$, $\chi_d$ varies over quadratic characters attached to fundamental discriminants $|d|\leq x$, $\lambda_f(n)$ are the Fourier coefficients of holomorphic cusp forms $f$ of (a large) weight $k$ for the full modular group, and $\text{Kl}_q(a, b)$ are the normalized Kloosterman sums modulo a large prime $q$, where $a, b$ vary in $(\mathbb{F}_q)^{\times}$.