Sums of four and more unit fractions and approximate parametrizations

Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equ…

奇异模迹的有效估计

Effective estimates for traces of singular moduli

本文针对奇异模的广义扭曲迹，利用 $Γ_0(4)$ 上 Kloosterman 和之和的显式界，建立了具有有效误差界的渐近公式，改进了 Andersen 和 Duke 的相关估计。

On exponential sums over orbits in $\mathbb{F}_p^d$

This paper proves a bound for exponential sums over orbits of vectors in $\mathbb{F}_p^d$ under subgroups of $\rm{GL}_d(\mathbb{F}_p)$. The main tool is a classification theorem for approximate groups due to Gill, Helfgott, Pyber, and Szabó.

On the theorem of Davenport and generalized Dedekind sums

A symmetrized lattice of $2n$ points in terms of an irrational real number $α$ is considered in the unit square, as in the theorem of Davenport. If $α$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be $c( α) \log n + O \left( \log \log n \right)$ for a computable positive constant $c( α)$. For the golden ratio $φ$, the value $\sqrt{c (φ) \log n}$ yields the smallest $L^2$ discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients $a_k$ of $α$ grow at most polynomially fast, the $L^2$ discrepancy is found in …

新形式戴德金和的大值

Large Values of Newform Dedekind Sums

本文研究新形式艾森斯坦级数上的广义戴德金和，证明其很少显著大于 $\log^3 c$，并通过连分数方法与特殊情形下的数值计算给出大值实例。

Approximate Pythagoras Numbers on $*$-algebras over $\mathbb{C}$

The Pythagoras number of a sum of squares is the shortest length among its sums of squares representations. In many algebras, for example real polynomial algebras in two or more variables, there exists no upper bound on the Pythagoras number for all sums of squares. In this paper, we study how Pythagoras numbers in $*$-algebras over $\mathbb{C}$ behave with respect to small perturbations of elements. More precisely, the approximate Pythagoras number of an element is the smallest Pythagoras number among all elements in its $\varepsilon$-ball. We show that these approximate Pythagoras numbers ar…

和集中缺失和的分布

Distribution of Missing Sums in Sumsets

研究和集缺失和的分布，推导方差公式并发现双峰分布特性。

关于广义Stirling数及特殊函数

On generalized Stirling numbers and special functions

本文提出第二类Stirling数的新推广，分析其性质并用于逼近Riemann zeta值，建立与特殊函数联系，并扩展到q-Stirling数，揭示组合与数论新见解。

Small Gaps Between Primes I

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes.

算术级数中素数幂的和

Sums of Powers of Primes in Arithmetic Progression

该研究将Gerard和Washington关于素数幂和近似素数计数的方法推广到算术级数中的素数，证明其渐近等价性，并量化了正负指数下近似值的偏差。

Every finite set of integers is an asymptotic approximate group

A set $A$ is an $(r,\ell)$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq X+A$. The set $A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an $(r,\ell)$-approximate group for all sufficiently large integers $h$. It is proved that every finite set of integers is an asymptotic $(r,r+1)$-approximate group for every integer $r \geq 2$.

On a Class of Sums with Unexpectedly High Cancellation, and its Applications

Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to an unexpected cancellation of oscillating sums. After stating the motivation, and our theorem, we apply it to prove several results on the Prouhet-Tarry-Escott Problem, integer partitions, and the distribution of prime numbers. Regarding the Prouhet-Tarry-Escott problem, we show that \begin{align*} \sum_{|\ell|\leq x}(4x^2-4\ell^2)^{2r}-\sum_{|\ell|<x}(4x^2-(2\ell+1)^2)^{2r}=\text{polynomial w.r.t. } x \text{ with degree }2r-1. \end{align*} This can perhaps be …