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

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

In this paper, we establish a large sieve inequality of Elliott-Montgomery-Vaughan type for Maass forms on GL(n, R) and explore three applications.

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

Let $a, b,c $ and $k$ be positive integers such that $1\leq a\leq b,a<c<2(a+b), c\ne b$ and $(a,b,c)=1$. Define the arithmetic function $f_k(a,b;c;n)$ by $$ \sum_{n=1}^{\infty}\frac{f_k(a,b;c;n)}{n^s}=\frac{ζ(as)ζ(bs)}{ζ^k(cs)}, \Re s >1.$$ Let $Δ_k(a,b;c;x)$ denote the error term of the summatory function of the function $f_k(a,b;c;n).$ IN this paper we shall give two expressions of $Δ_k(a,b;c;x)$. As applications, we study the so-called $(l,r)$-integers, the generalized square-full integers, the $e-r$-free integers, the divisor problem over $r$-free integers, the $e$-square-free integers. An…

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.

Distribution of Values of Symmetric power L-functions at the Edge of the Critical Strip

We study some problems on the distribution of values of symmetric power $L$-functions at $s=1$ in both aspects of level and weight: bounds of these values, extreme values, Montgomery-Vaughan's conjecture and distribution functions. Our results generalize and/or improve related results of Royer-Wu, Cogdell-Michel, Lau-Wu and Liu-Royer-Wu.

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。

On sparsity of positive-definite automorphic forms within a family

It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. On generalizing their result, we formulate an axiomatic result about sets of automorphic forms satisfying certain averages when suitably ordered, which ensures that almost all forms are not positive-definite within such sets. We then apply the result to various families, including the famil…