Möbius函数与多项式的正交性及其在$\mathbb{F}_p[x]$中素数线性方程的应用

M{ö}bius orthogonality in density for zero entropy dynamical systems

It is proved that whenever a zero entropy dynamical system $(X,T)$ has only countably many ergodic measures and $μ$ stands for the arithmetic M{ö}bius function, then there exists a subset $A$ of integers depending only on the system, of logarithmic density one, such that for each $f$ continuous on $X$, $\frac1N \sum_{n\leq N} f(T^nx)μ(n) \to 0$ as $N\to\infty$, $N\in A$, uniformly in $x\in X$. In particular, the density version of M{ö}bius orthogonality holds.

关于素数上的某些求和与里斯函数

On certain sums over primes and the Riesz function

本文通过Gram级数导数与里斯函数的关系，探讨素数求和近似中的Mobius函数级数，推导出一般积分关系。

Reciprocity and orthogonality

Let $p$ be a prime and let $K$ be a finite extension of the field ${\bf Q}_p$ of $p$-adic numbers such that the group ${}_pK^\times$ has order $p$. The ${\bf F}_p$-space $K^\times\!/K^{\times p}$ carries a natural filtration coming from the valuation on $K$, and a natural bilinear pairing coming from the reciprocity isomorphism for the exponent~$p$. We determine the orthogonal filtration for this pairing. We also prove the analogous result for $p$-fields of characteristic $p$.

半整数权模形式的Selberg正交性

Selberg orthogonality for half-integral weight modular forms

本文基于半整数权模形式的Selberg正交性，利用GRH和广义拉马努金猜想，推导了三个算术应用的去相关性结果。

GL(3)系数与线性及非线性相位加法特征之间的正交关系

Orthogonality relation between $GL(3)$ coefficients and additive characters with linear and non-linear phases

该研究针对SL(3,Z)的Hecke-Maass形式的素数支撑傅里叶系数，通过构造Vaughan恒等式自守变体、改进双线性型等方法，突破了Jiang与吕的2/3障碍，拓宽了非平凡抵消的容许范围。

${\rm GL}(n, \mathbb R)$ 的一个渐近正交关系

An Asymptotic Orthogonality Relation for ${\rm GL}(n, \mathbb R)$

本文将 GL(4,R) 的显式渐近正交关系推广至 GL(n,R) (n≥2)，利用 Kuznetsov 迹公式等新技术，在 n≤5 时给出无条件结果，n>5 时基于两个猜想得出结论。