A large sieve inequality of Elliott--Montgomery--Vaughan type for Maass forms on GL(n,R) with applications

关于自同构L函数在1处的极值的Montgomery-Vaughan猜想

On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1

本文证明了Montgomery-Vaughan关于自守L函数在1处的极值的一个猜想的一个较弱形式。

一类算术函数的条件结果：H. L. Montgomery和R. C.沃恩法

Conditional Results for a Class of Arithmetic Functions: a variant of H. L. Montgomery and R. C. Vaughan's method

令$a, b,c $和$k$是正整数，使得$1\leq a\leq b,a<c<2(a+b), c\ne b$和$(a,b,c)=1$。用$$ \sum_{n=1}^{\infty}\frac{f_k(a,b;c;n)}{n^s}=\frac{ζ(as)ζ(bs)}{ζ^k(cs)}, \Re s >1.$$定义算术函数$f_k(a,b;c;n)$。令$Δ_k(a,b;c;x)$表示函数$f_k(a,b;c;n).$的和函数的误差项。本文给出了$Δ_k(a,b;c;x)$的两种表达式。作为应用，我们研究了所谓的$(l,r)$ -整数，广义平方满整数，无$e-r$整数，无$e-r$整数上的除数问题，无$e$ -平方整数.一个重要的工具是推广的方法H。L. Montgomery和R. C.沃恩

A Montgomery-Hooley theorem for the k-fold divisor function

Let $d_k(n)$ denote the $k$-fold divisor function. For a wide range of large $q$ the expected bound $$\sum_{n\leq x\atop {n\equiv a(q)}}d_k(n)-\text { main term }\approx \sqrt {x/q}$$ is shown to be true in an average sense -- for all $k$. This generalises the work of Pongsriiam and Vaughan [15] who studied $k=2$, and answers the work of Rodgers and Soundararajan [17], who used the asymptotic large sieve to study a smoothed version of the problem. We use a circle method approach as developed by Goldston and Vaughan [7] to study the unsmoothed problem.

Dirichlet L-函数零点的配对相关性：通往Chowla、Elliott-Halberstam和Montgomery猜想的可能路径

Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery

在假设广义黎曼猜想和Dirichlet L-函数零点的配对猜想下，证明了Montgomery关于算术进展中素数定理误差项大小的猜想，并推导出Elliott-Halberstam猜想成立，以及Dirichlet特征L(1/2,χ)=0的数量小于q^{1/2+ε}。

带乘性系数的Weyl和与联合等分布

Weyl sums with multiplicative coefficients and joint equidistribution

该研究将Montgomery与Vaughan的带乘性系数指数和结果推广至Weyl和，得到多项式同余根与值的联合等分布结论及混合特征和新结果。

具有乘法系数的指数和的有效估计

Effective estimates for exponential sums with multiplicative coefficients

本文改进了具有乘法系数的指数和估计，去除了Bachman结果中的$\sqrt{\log R}$因子，同时保留Montgomery-Vaughan的原始假设，并证明平方根位移依赖的锐性。