Stephan Baier, Liangyi Zhao
In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form $p=am^2+1$ such that $a\leq p^{5/9+ε}$.
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A Bombieri-Vinogradov theorem for sectors in real quadratic number fields
本文在实二次数域中建立扇区的Bombieri-Vinogradov定理，研究数论中的分布问题。
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
Let $f$ and $g$ be $1$-bounded multiplicative functions for which $f*g=1_{.=1}$. The Bombieri-Vinogradov Theorem holds for both $f$ and $g$ if and only if the Siegel-Walfisz criterion holds for both $f$ and $g$, and the Bombieri-Vinogradov Theorem holds for $f$ restricted to the primes.
In this paper we establish a generalization of Bombieri-Vinogradov theorem for primes represented by a fixed positive definite binary quadratic form. Then we apply this theorem to generalize a result of Vatwani on bounded gap between Gaussian primes.
The Bombieri-Vinogradov theorem for primes of the form $\mathbf{p=x^2+y^2+1}$
该研究针对形如$p=x^2+y^2+1$的素数，借助包络筛法证明了对应的Bombieri-Vinogradov型结果，丰富了特殊素数的分布理论。
We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem to this setting, we apply the polynomial large sieve inequality that has been recently proved and which includes recent progress in Vinogradov's mean value theorem due to Parsell et al. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.
If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.
In this paper we improve the result of Akbary and Hambrook by a factor of log x by obtaining a better version of Vaughan's inequality and using the explicit variant of an inequality connected to the Möbius function, derived by Helfgott in his work on ternary Goldbach conjecture.
We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λand a Dirichlet character χ. As a consequence of this result we prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.
We prove Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for the y-smooth numbers less than x, on the range log^{K}x \leq y \leq x. This improves on the range \exp{log^{2/3 + ε}x} \leq y \leq x that was previously available. Our proofs combine zero-density methods with direct applications of the large sieve, which seems to be an essential feature and allows us to cope with the sparseness of the smooth numbers. We also obtain improved individual (i.e. not averaged) estimates for character sums over smooth numbers.
A Bombieri-Vinogradov-type theorem for moduli with small radical
本文将关于素数幂稀疏集的 Bombieri-Vinogradov 型定理推广至具有小根的模的稀疏集，通过结合先前方法与 Roger Baker 的一般模结论得出结果。
We establish results of Bombieri-Vinogradov type for the von Mangoldt function $Λ(n)$ twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase $e(P(n))$; the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes $p$ obeying a "nil-Bohr set" condition, such as $\|αp^k\|<\varepsilon$, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr…
A Bombieri-Vinogradov Theorem for primes in short intervals and small sectors
本文研究了在短区间和小扇区条件下的素数分布，证明了其在算术级数中的良好分布性，扩展了Duke和Coleman之前的成果。
Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_{q}$. For arithmetic functions $ψ_{1}, ψ_{2}: \mathbb{F}_{q}[t]\rightarrow\mathbb{C}$, we establish that if a Bombieri-Vinogradov type equidistribution result holds for $ψ_{1}$ and $ψ_{2}$, then it also holds for their Dirichlet convolution $ψ_{1} \ast ψ_{2}$. As an application of this, we resolve a version of the Titchmarsh divisor problem in $\mathbb{F}_{q}[t]$. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in $\mathbb{F}_q[t]$.
Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limita…
A Bombieri-Vinogradov theorem for exponential sums over products of k primes
本文提出了一种关于素数乘积上指数和的Bombieri-Vinogradov定理，并给出了其在数论中的应用及下界估计。
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。
We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are "twisted" by a splitting condition in a Galois extension $L/K$ of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes in short intervals with a given splitting condition in a Galois extension $L/\mathbb{Q}$ exhibit dense clusters in short intervals. We explore several arithmetic applications related to questions of Se…
An effective Bombieri-Vinogradov error term for sifting problems
本文通过改进筛法上下界，避免了Siegel零的存在问题，证明了经典筛问题可有效化，同时给出了如素数p≤x且p+2也为素数的计数上界等应用。
A Bombieri-Vinogradov theorem for higher rank groups
该研究针对数域上非自二次扭的GL_n自对偶尖点自守表示，建立了素理想处Dirichlet系数的无条件Bombieri-Vinogradov型定理，改进了依赖未证假设的分布水平，并给出低秩情形的数论应用。
In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.
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…
We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.
Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications
本文将带良因子可分权重的Bombieri-Vinogradov定理推广为卷积形式，改进了Wang的相关结果，并在最大素因子相关的素数分布问题中得到了更优的渐近密度上界。
We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.
We shall give an explicit version of Bombieri-Vinogradov Theorem for moduli not divisible by an exceptional modulus.
Bounded gaps between primes
Polymath8b 团队证明素数相邻差的下确界不超过246，本文结合 Bombieri-Vinogradov 定理与光滑模等分布估计，将该边界优化至不超过240。
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.
We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $τ(n-1)$ where $τ$ denotes the divisor function.
We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.
We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square free moduli and also on the result due to Gallagher and Iwaniec when the core $q'=\prod_{p|q}p$ of the modulus $q$ satisfies $\log q'\sim \log q$. Some applications to zero free regions of Dirichlet L-functions and the $\rm{P\acute{o}lya}$ and Vinogradov inequalities are indicated.
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+δ}$ with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but $O(δQ)$ moduli $q\sim Q=x^{1/2+δ}$ and we get results for moduli as large as $x^{11/21}$. Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and P…
Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory
本文利用圆法证明了缺失一位数字的素数的Bombieri-Vinogradov型定理，并结合半线性筛，给出了形如p=1+m^2+n^2的此类素数数量的正确数量级上下界。
The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of…
Primes in arithmetic progressions to large moduli and refinements of Harman's sieve
利用多作者算术信息与Harman筛法变体，构造素数指示函数的上下界，得到模数达x^{9/17}的双线性形式和模数达x^{17/32}的三线性形式的均值定理，并给出π(x;q,a)的几乎全部模数q的新上下界。
We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K.~Halupczok (2012, 2015, 2018) and M.~Munsch (2020). We also consider moduli defined by polynomials $f(X) \in \mathbb{Z}[X]$, Piatetski-Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri--…
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。
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。
On Chen's theorem over Piatetski-Shapiro type primes and almost-primes
本文在Piatetski-Shapiro序列上建立新的Bombieri-Vinogradov型均值定理，推广并改进了先前关于该序列上素数、殆素数的陈氏定理相关结果。
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