The Erdős discrepancy problem over the squarefree and cubefree integers

无幂整数与傅里叶界

Powerfree integers and Fourier bounds

本文提出一种不依赖等分布估计的通用方法，利用傅里叶变换的$L^{\infty}$、$L^{1}$界及递减性证明整数集含无穷多个无幂数，并应用于回文数、倒序数及缺数字数，得到新界与无穷性结果。

带缺失数位的无平方因子多项式

Squarefree polynomials with missing digits

该研究针对系数取自{0,1}的有限域上的多项式，建立其为无平方因子的渐近公式，解决了相关猜想的函数域类似问题，并估计了特定随机多项式为无平方因子的概率。

On a problem of Hoffstein and Kontorovich

Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ and $d$ be a fundamental discriminant. Hoffstein and Kontorovich ask for a bound on the least $|d|$ (if it exists) such that the central value $L(1/2, π\otimes χ_d) \neq 0$. The bound should be given in terms of the weight, Laplace eigenvalue and/or level of $π$. Let $f$ be a holomorphic twist-minimal newform of even weight $\ell$, odd cubefree level $N$, and trivial nebentypus. When $π\cong π_f$ and the squarefree part of $N$ is of appropriate size, we conditionally improve upon level aspect res…

关于无平方因子整数与2的幂之和

On the Sum of Squarefree Integers and a Power of Two

本研究针对Erdos提出的“大于1的奇数均可表为无平方因子数与2的幂之和”的猜想，采用GPU上的高度并行化算法，将数值验证范围拓展至$2^{50}$以内的所有奇数，较此前结果提升超80万倍，同时给出了相关新的数值发现。

关于平方自由数和平方满数的一些遍历定理

Some ergodic theorems over squarefree numbers and squarefull numbers

本文基于平方自由数和平方满数的不变平均性质，推导出若干遍历定理，与Erdős-Kac定理、Bergelson-Richter定理及Loyd定理有重要关联。

On the Bivariate Erdős-Kac Theorem and Correlations of the Möbius Function

Let $2 \leq y \leq x$ such that $β:= \frac{\log x}{\log y} \rightarrow \infty$. Let $ω_y(n)$ denote the number of distinct prime factors $p$ of $n$ such that $p \leq y$, and let $μ_y(n) := μ^2(n)(-1)^{ω_y(n)}$, where $μ$ is the Möbius function. We prove that if $β$ is not too large (in terms of $x$) then for each fixed $a \in \mathbb{N}$, \begin{equation*} \sum_{n \leq x} μ_y(n)μ_y(n+a) \ll x\left(\frac{1}{\log_2 y} + e^{-\frac{1}{21}β\log β}\right). \end{equation*} This can be seen as a partial result towards the binary Chowla conjecture. Our main input is a \emph{quantitative} bivariate anal…