关于光滑数的Bennieri-Vinogradov和Barban-Davenport-Halbernovel型定理

稀疏模集上的Bennieri-Vinogradov型定理

Bombieri-Vinogradov Type Theorem for Sparse Sets of Moduli

本文建立了模的稀疏集的Bennieri-Vinogradov型定理和Barban-Davenport-Halbergov型定理。作为应用，我们证明了存在无穷多个$p=am^2+1$形式的素数使得$a\leq p^{5/9+ε}$ .

关于等差数列中$k-$自由数孪生的方差

On a variance for twins of $k-$free numbers in arithmetic progressions

本文给出了等差数列中$k-$自由数的孪生数的Barban-Davenport-Halbernut型的一个新的上界.

Twins almost prime under a Elliott-Halberstam's conjecture

We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all $\varepsilon>0$, $p-2$ is either a prime number or can be written as $p_1 p_2$ where $p_1$ and $p_2$ are prime and $p_1<X^{\varepsilon}$, and we give the explicit asymptotic.

An average theorem for tuples of k-free numbers in arithmetic progressions

We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.

Bombieri-Vinogradov定理的对数改进

A logarithmic improvement in the Bombieri-Vinogradov theorem

本文利用Vaughan恒等式的加权形式及Selberg筛法相关估计，将Bombieri-Vinogradov定理的误差项从(log x)^(5/2)改进至(log x)^2，并给出了有效与非有效版本。

阿贝尔数域中的小素隙

Small Prime Gaps in Abelian Number Fields

我们证明了一个类似的结果Goldston，Pintz和Yildirim的素数之间的小差距，完全分裂在一个阿贝尔数域。我们证明了一个条件的结果假设的Elliott-Halbertmann猜想，和一个无条件的结果。在类数1的二次扩张的特殊情况下，我们也给出了相同结果的另一个证明，它依赖于二次数域上的Bennieri-Vinogradov定理的推广。

A lower bound for the variance in arithmetic progressions of some multiplicative functions close to $1$

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $α_N$-fold divisor functions, when $α_N$ is a sequence of positive real numbers approaching $1$ in a suitable way or $α_N=1$, and the indicator of $y$-smooth numbers, for suitably large parameters $y$. As a corollary, we will strengthen a previous author's result on the first subject and obtain matching lower bounds to some Barban-Davenport-Halberstam type theorems for $y$-smooth numbers. Incidentally, we will also find a lower bound for the vari…

A Bombieri-Vinogradov-type theorem with prime power moduli

In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose cardinality is bounded by a power of $\log x$. We use a $p$-adic variant Harman's sieve to extend Baker's range to $q\le x^{1/4-\varepsilon}$ if $\mathcal{S}$ is restricted to prime powers $p^N$, where $p\le (\log x)^C$ for some fixed but arb…

无平方因子级全纯新形式的联合水平 - 权重波动定律

Joint level-weight murmurations: prime averaging and the cubic pointwise range

研究为无平方因子级全纯新形式建立联合水平 - 权重波动定律，从精确迹公式出发，经权重求和、截断\(L\)值、平均周期函数及诺伊曼求和等步骤，无需\(L\)函数零点假设，得到收敛到特定测度的结果。

关于Elliott的一个Erdős--Kac型猜想

On an Erdős--Kac-type conjecture of Elliott

本文证明了Elliott基于Erdős--Kac定理提出的一个关于素数求和渐近性的猜想可由Bombieri--Vinogradov定理推出，并利用素数更深层的分布水平结果证明了一个涉及Poisson--Dirichlet分布的相关结论。

短算术级数中的积性函数

Multiplicative functions in short arithmetic progressions

该研究探讨有界积性函数在短算术级数中的和的方差，改进推广了相关经典定理，给出近乎最优的例外模集合，并将结果应用于三素数乘积的Linnik型问题。