Hongze Li, Hao Pan
Let P denote the set of all primes. Suppose that P_1, P_2, P_3 are three subsets of P with the sum of their lower densities relative to P is greater than 2. We prove that for sufficiently large odd integer n, there exist p_i\in P_i such that n=p_1+p_2+p_3.
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In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
In this paper, we proved a theorem that every large enough odd number can be represented as the sum of three almost equal Piatetski-Shapiro primes.
Vinogradov's three primes theorem in the intersection of multiple Piatetski-Shapiro sets
本文将维诺格拉多夫三素数定理推广到多个Piatetski-Shapiro序列的交集情形，证明充分大奇数仍可表示为三个来自该交集的素数之和。
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。
In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, the interval $[p_i, p_i + H]$ contains at least $m$ primes, for some $H = H(m)$. Second, every sufficiently large integer $N \equiv 3 \pmod{6}$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, $p_i + 2$ has at most two prime factors.
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.
We show that every sufficiently large $x\equiv 3(4)$ can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas.
We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible.
We prove a generalisation of Vinogradov's theorem by finding for $m\geqslant 3$ and fixed positive integers $c_1, \dots ,c_m, r_1, \dots , r_m$ the asymptotics of the number of sequences $(n_1, \dots ,n_m) \in \mathbf{N}^{m}$ such that $c_1n_1 + \dots + c_m n_m = N$ and $Ω(n_i) = r_i$ for every $i=1, \dots ,m$ under the assumption that at least three of the $r_i$ are equal to $1$.
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。
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 first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.
In this paper we proved a new numerically explicit version of the Pólya--Vinogradov inequality. Our proof is based on the new ideas of V.A. Bykovskii and improves a recent inequality obtained by C. Pomerance.
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。
Vinogradov's theorem for primes with restricted digits
本文在基 g 展开中不含某固定数字的整数集合上，证明了每个充分大的奇数都能表示为三个此类素数之和，从而将维诺格拉多夫三素数定理推广到受限数字的素数情形。
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…
In this paper, if prime $p\equiv 3\pmod 4$ is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length $k$ as $k$ tends to $\lceil \log_2 p\rceil$. As an immediate consequence, it proves that, there exist a constant $c$ such that, the least nonresidue for such primes is at most $c\lceil \log_2 p\rceil$.
We give a new proof of Vinogradov's three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of L-functions, either explicitly or implicitly. Our proof uses instead a transference principle, the idea of which was first developed by Green. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.
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 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 $a\leq p^{5/9+ε}$.
Let $θ> 11/20$. We prove that every sufficiently large odd integer $n$ can be written as a sum of three primes $n = p_1 + p_2 + p_3$ with $|p_i - n/3| \leq n^θ$ for $i\in\{1,2,3\}$.
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.
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…
We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.
We present an assessment of the distance in total variation of \textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]=\{1,\dots, n\}$ and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius et al. which consider the particular case of uniform samples in $[n]$ and co…
Prime Solutions of Diagonal Diophantine Systems
该研究通过建立对角丢番图方程组素数解的渐近公式，结合辛格罗多夫定理和霍利工作，给出了沃林-戈尔德巴奇问题的条件结果。
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.
For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold for almost all primes $p$ but remains open in general. In this paper we show that Vinogradov's conjecture also follows from the Elliott-Halberstam conjecture on the distribution of primes in arithmetic progressions, thus providing a potential "non-multiplicative" route to the Vinogradov conjecture. We also give a variant of this argument that obtains bounds …
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。
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 establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.
We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.
We make explicit a theorem of Fromm and Goldmakher [arXiv:1706.03002], which states that one can improve Burgess' bound for short character sums simply by improving the leading constant in the Pólya-Vinogradov inequality. Towards achieving this, we establish explicit versions of several estimates related to the mean values of real multiplicative functions and the Dickman function.
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.
Vinogradov three prime theorem with Piatetski-Shapiro primes
本文证明当指数在(1,41/35)内时，每个充分大的奇数都可表示为三个Piatetski-Shapiro素数之和，方法结合Green转移原理变体、Bourgain限制性估计与Harman筛法。
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 $\mathbb{F}_q[t]$.
We shall give an explicit version of Bombieri-Vinogradov Theorem for moduli not divisible by an exceptional modulus.
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 give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized values. This will give an improved explicit Bombieri-Vinogradov like result for non-exceptional moduli.
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…
We use Vinogradov's method to prove equidistribution of a spin symbol governing the $16$-rank of class groups of quadratic number fields $\mathbb{Q}(\sqrt{-2p})$, where $p \equiv 1 \bmod 4$ is a prime.
Let $α> 0$ be any fixed non-integer, $I$ be any subinterval of $[0; 1)$. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the set of primes $p$ satisfying the condition $\{ p^α \} \in I$. This strengthens the previous result of Gritsenko and Zinchenko.
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 continue to study the distribution of prime numbers $p$, satisfying the condition $\{ p^α \} \in I \subset [0; 1)$, in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for $0 < α< 1/9$ with the level of distribution $θ= 2/5 - (3/5) α$, which improves the previous result corresponding to $θ\leq 1/3$.
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.
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/γ}]$ with $\frac{85}{86}<γ<1$. Moreover, we use this result to prove that, for $0.9989445<γ<1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2=\mathcal{P}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.
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。
For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is equal to $1/8$, in line with a more general conjecture of Gerth. Vinogradov's method is the key analytic tool for our work.
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。
We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining polynomials of some pre-homogeneous vector spaces. Three of these families are instances of a conjecture by Bourgain, Gamburd and Sarnak regarding prime points in orbits of simple algebraic groups. Our approach is based on the formulation of a general condition on the defining polynomial of a hypersurface, which suffices to guarantee that Zariski-density of…
Let $p$ be an odd prime. Using I. M. Vinogradov's bilinear estimate, we present an elementary approach to estimate nontrivially the character sum $$ \sum_{x\in H}χ(x+a),\qquad a\in\Bbb F_p^*, $$ where $H<\Bbb F_p^*$ is a multiplicative subgroup in finite prime field $\Bbb F_p$. Some interesting mean-value estimates are also provided.
We study the prime numbers that lie in Beatty sequences of the form $\lfloor αn + β\rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $α^{-1}\cdot\frac{|C|}{|G|}$. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri--Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bound…
Some mean value theorems in the style of Bombieri-Vinogradov's theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvable with primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] with X_{i}^{θ_i}=Y, i=1,2, and θ_1,θ_2\geq 0.933 such that (p_1+2)(p_2+2) has at most 9 prime factors.
Refinements to the prime number theorem for arithmetic progressions
本文证明了等差数列素数定理的一致版本，并由此推出多个经典结果，证明依赖于Vinogradov-Korobov无零点区域、无对数零点密度估计和Deuring-Heilbronn零点排斥现象。
Given a family $φ= (φ_1, \ldots, φ_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $ρ$, we consider the mean value $$ M_{k, ρ} (φ, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ(\mathbf{x}, \mathbf{y}; N) \right|^ρd\mathbf{x} $$ of exponential sums $$ S_φ( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 πi \left(\sum_{j=1}^k x_j φ_j(n)+ \sum_{j=1}^{d-k}y_jφ_{k+j}(n)\right)\right), $$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of poly…
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 prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Polya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements.
In the recent preprint [3], Goldston, Pintz, and Yıldırım established, among other things, $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0,\leqno(0) $$ with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. While [3] also includes quantitative versions of $(0)$, we are concerned here solely with proving the qualitative $(0)$, which still exhibits all the essentials of the method. We also show here that an improvement of the Bombieri--Vinogradov prime number theorem would give rise infinitely …
The Exceptional Set in Goldbach's Problem with two Chen Primes
该研究针对满足$n\equiv4\pmod{6}$的自然数，通过结合Bombieri–Vinogradov定理的幂节省变体的高效筛策略，改进了哥德巴赫问题中两个陈素数和的相关结论，确定了其例外集的幂节省性质。
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。
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