Spectral decomposition formula and moments of symmetric square L-functions

Continuants and some decompositions into squares

In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number $p$ as a sum of two squares, given a solution of $z^2+1\equiv0\pmod{p}$, and vice versa. In this paper, we extend the use of continuants to proper representations by sums of two squares in rings of polynomials on fields of characteristic different from 2. New deterministic algorithms for finding the corresponding proper representations are presented. Our approach will provide a new constructive proof of the four-square theorem …

大筛法不等式与平方和的加法分解

The large sieve inequality and additive decompositions of sums of squares

该研究基于大筛法证明了针对缺失素数模平方剩余类集合的新筛法估计，将其应用于Ostmann问题，得出平方和的非平凡和集分解需由大小相近的两个集合构成的结论。

利用类群进行整数的快速无平方因子分解

Fast square-free decomposition of integers using class groups

本文提出一种基于二元二次型类群的算法，用于快速求 $n=a^2b$ 的无平方因子分解，在 $a,b$ 为密码学规模素数时是目前最快的方法，对依赖此类整数分解困难性的密码系统具有重要意义。

On squares in subsets of finite fields with restrictions on coefficients of basis decomposition

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let ${a_1,\ldots,a_r}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique representation in the form $\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$. Let $D$ be a subset of $\mathbb{F}_p$. We consider the set $W_D$ of elements of $\mathbb{F}_q$ such that $c_j\in D$ for all $j=1,\ldots,r$. We give an estimate for the number of squares in the set $W_D$ which implies a weaker sufficient condition for the existence of squares in the set $W_D$…

Decomposition of the Catalan number into the sum of squares

Analysis of the dynamics of the Dyck words helped solve the problem of representing the Catalan number as a sum of squares of natural numbers. In this case, the Dyck triangle is considered in different coordinates. In the calculations, we use the Catalan convolution matrix.

The first moment of Maass form symmetric square L-functions

We prove an asymptotic formula for the twisted first moment of Maass form symmetric square L-functions on the critical line and at the critical point. The error term is estimated uniformly with respect to all parameters.