In this paper, we investigate the large deviations of sums of weighted random variables that are approximately independent, generalizing and improving some results of Montgomery and Odlyzko. We are motivated by examples arising from number theory, including the sequences p^it, χ(p), χ_d(p), λ_f (p), and Kl_q(a − n, b), where p ranges over the primes, t varies in a large interval, χ varies among all characters modulo q, χ_d varies over quadratic characters attached to fundamental discriminants |d| ≤ x, λ_f (n) are the Fourier coefficients of holomorphic cusp forms f of (a large) weight k for the full modular group, and Kl_q(a, b) are the normalized Kloosterman sums modulo a large prime q, where a, b vary in (𝔽_q)^×.

在本文中，我们研究了近似独立的加权随机变量和的大偏差，概括和改进了 Montgomery 和 Odlyzko 的一些结果。我们受到数论中的例子的启发，包括序列p ^{i t} , χ ( p ), χ _d ( p ), λ _f ( p ) 和 Kl _q ( a − n, b )，其中p 的范围是素数, t在一个很大的区间内变化, χ在所有字符模q之间变化,χ _d随附于基本判别式 |d| 的二次特征而变化 ≤ x , λ _f ( n ) 是全模群的（大）权重k的全纯尖点形式f的傅立叶系数，Kl _q ( a, b ) 是模大素数q的归一化 Kloosterman 和，其中a , b在 (𝔽 _q ) ^{× 变化}。