A Conditional Explicit Result for the Prime Number Theorem in Short Intervals

Sums of one prime power and two squares of primes in short intervals

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.

包含至少三个素数的极小区间

Very small intervals containing at least three primes

设$p_n$是第$n$个素数。借助于Cramér-like模型，我们证明了含有3个素数的$(2p_n,\enskip2p_{n+1})$形式的区间集关于所有这种形式的区间集有正密度.

关于短区间中素数指数和的一个条件性结果

A conditional result on exponential sums over primes in short intervals

本文在广义 Riemann 假设及 Dirichlet L 函数密度假设同时成立的条件下，给出短区间中素数指数和的一个条件性结果。

短区间素数与素数上指数和的显式关系

Explicit relations between primes in short intervals and exponential sums over primes

在黎曼假设（RH）的假设下，证明了素数上指数和的截断均方平均渐近公式中的假设误差项与短区间内素数的均方误差项之间的定量关系.我们还注意到，这样的关系与更精确的形式蒙哥马利的配对相关猜想。

短区间素数的高阶矩I

Higher moments of primes in short intervals I

本文在黎曼假设下证明了短区间内素数的两个高偶矩之间的一个“等价性”。我们还提供了数值证据支持这些渐近公式。

包含两个素数和一个准素数的有界长区间

Bounded length intervals containing two primes and an almost-prime

Goldston、Pintz和Yıldırım证明了，如果素数对于某个$θ>1/2$有“分布水平”$θ$，那么存在一个常数$C(θ)$，使得有无穷多个整数$n$，其中$[n,n+C(θ)]$区间包含两个素数。在同样的假设下，我们证明了对于任意整数$k\ge 1$，存在常数$D(θ,k)$和$r(θ,k)$，使得有无穷多个整数$n$，其中区间$[n,n+D(θ,k)]$包含两个素数和$k$几乎素数，且所有几乎素数至多有$r(θ,k)$素因子.如果$θ$可以取为与$1-ε$一样大，并且假设具有2个、3个或4个素因子的数也具有分布水平$1-ε$，则我们证明存在无限多个整数$n$，使得区间$[n,n+90]$包含2个素数和一个最多具有4个素因子的数。

Conditional upper bound for the k-th prime ideal with given Artin symbol

We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.

短区间上素数的高阶矩II

Higher moments of primes in short intervals II

在充分了解偶数阶矩的基础上，导出了素数在短区间内的$λ$阶矩的渐近公式，并证明了奇阶矩的“等价性”。我们还提供了支持这些结果的数字证据。

算术等差数列中最小素数的条件界

A conditional bound for the least prime in an arithmetic progression

在假设狄利克雷L函数广义 Lindelöf 假设下，研究了算术等差数列中最小素数的界，其结果接近广义黎曼假设所暗示的经典估计。

Bounded gaps between primes in short intervals

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $δ\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^δ,x]$ contains $\gg_{k} \frac{x^δ}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be re…

Short intervals asymptotic formulae for binary problems with primes and powers, II: density $1$

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.

Short interval results for certain prime-independent multiplicative functions

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.

关于包含素数的临界小区间

On critical small intervals containing primes

设p是奇素数，使得p_n<p/2< p_{n+1}，其中p_n是第n个素数。本文研究如下问题：在区间（p，2 p_{n+1}）中素数存在的概率是多少？在借助Ramanujan素数（[11]，[12]）和引入伪Ramanujan素数强定义概率之后，我们证明了，如果这样的概率P存在，则P>=0.5。我们还研究了一个对称的情况下，左间隔，它连接到序列A080359在OEIS。

短区间内素数和与素数平方和

Sums of primes and squares of primes in short intervals

令$\mathbf H_2$表示偶数集合$n \not\equiv 1 \pmod 3$。我们证明了当$H \ge X^{0.33}$，几乎所有的整数$n \in \mathbf H_2$，$X < n \le X + H$都可以表示为一个素数与一个素数的平方之和.我们也证明了一个类似的结果，三个平方和的素数。

Short intervals asymptotic formulae for binary problems with primes and powers, I: density $3/2$

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.

Primes in explicit short intervals on RH

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

Prime polynomial values of linear functions in short intervals

In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field.

短区间上的有界素数间隙

Bounded prime gaps in short intervals

本文推广了Zhang和Pintz关于有界素缺的最新结果，给出了短区间$[x,x+x (\log x)^{-A}]$中有界素对个数为6* 10 ^7的下界.我们的结果只通过分析张的定理1的证明，但我们也解释了如何张的定理2的一个尖锐的变体将意味着相同的结果为更短的时间间隔。

短区间和剩余类中素多项式个数的方差

The variance of the number of prime polynomials in short intervals and in residue classes

我们解决了一个函数域版本的两个apturtures关于方差的素数在短时间间隔（戈德斯通和蒙哥马利）和等差数列（胡利）。我们工作中的一个重要组成部分是最近N的等分布结果。卡茨

短区间内零与素数的更精确配对相关

More precise pair correlation of zeros and primes in short intervals

Goldston和Montgomery [3]在黎曼假设下证明了短区间内素数的强对相关猜想与两个二阶矩是等价的。在本文中，我们得到了上述每一个的第二个主要的条款，并表明他们几乎是相互等价的。

Primes in the intervals between primes squared

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of this essential characteristic and present evidence for the conjecture that $π_k \sim |s_k|/ \log p_{k+1}^2$, where $π_k$ is the number of primes in $s_k$; or even stricter, that $y=x^{1/2}$ is both necessary and sufficient for the prime number theorem to be valid in intervals of length $y$. In addition, we propose and substantiate that the prime counting fun…

An extended pair-correlation conjecture and primes in short intervals

In this paper we extend the well-known investigations of Montgomery and Goldston & Montgomery, concerning the pair-correlation function and its relations with the distribution of primes in short intervals, to a more general version of the pair-correlation function.

On the Prime Numbers in the Interval [4n,5n]

Is it true that for all $n\geq k\geq 2$ there exists a prime number between $kn$ and $(k+1)n$? In this paper we show that there is always a prime number between $4n$ and $5n$ for all $n>2$. We also show there are at least seven prime numbers between $n$ and $5n$ for all $n>5$.

关于短区间素数的注记

A note on Primes in Short Intervals

我们可以假设Hardy-Littlewood素数$k$ -元组猜想的平均形式，而不是强定量形式，并且仍然得到Montgomery和Soundarajan [1]关于$ψ(x+h) - ψ(x)$的相同分布结果.

Discorrelation between primes in short intervals and polynomial phases

Let $H = N^θ, θ> 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} Λ(n) e(g(n)), \] where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = αn^k$, we deduce a short interval version of the Waring-Goldbach problem.

关于素隙和甚短间隔的注记

Note On Prime Gaps And Very Short Intervals

本文在黎曼假设的前提下，讨论了一个新的初等论证，它似乎证明了一个有限序列的素数p_1，p_2，.的最大素数间隔，p_n <= x，满足max {p_（n+1）- p_n：p_n <= x} <= c1（（logx）^2）/loglogx，c1 > 0常数。等价地，它表明非常短的区间（x，x + y）包含素数，对于所有y > c2（（logx）^2）/loglogx，c2 > 0常数，并且x足够大> 0。

Bounded intervals containing many primes

By combining a sieve method of Harman with the work of Maynard and Tao we show that $$\liminf_{n\rightarrow \infty}(p_{n+m}-p_n)\ll \exp(3.815m).$$

Explicit Interval Estimates for Prime Numbers

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(Δ,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - Δ^{-1}), x]$.

A heuristic study of the distribution of primes in short and not-so-short intervals

A numerical study on the distributions of primes in short intervals of length $h$ over the natural numbers $N$ is presented. Based on Cramér's model in Number Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.

Consecutive primes in short intervals

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.

对数区间中的素数

Prime numbers in logarithmic intervals

设$X$为大参数。本文首先给出$(p,p+h]$型素数在短区间内的积分矩的一个新的估计，这里$p\leq X$是素数，$h=\odi{X}$是素数。然后，我们将应用这一点来证明，对于每个$λ>1/2$，存在一个正比例的素数$p\leq X$，使得区间$(p,p+ λ\log X]$至少包含一个素数。作为结果，我们改进了Cheer和Goldston关于实数$λ>1$的大小的结果，并证明了存在正比例的整数$m\leq X$使得区间$(m,m+ λ\log X]$不含素数.我们还证明了其他结果的时刻之间的差距连续素数和整数$m\leq X$的正比例，使间隔$(m,m+ λ\log X]$包含至少一个素数。这些技术的最后应用是两个定理（第一个是无条件的，第二个是我们假设黎曼假设和蒙哥马利对相关猜想的一种形式的有效性），关于素数$p\leq X$的正比例，使得区间$(p,p+ λ\log X]$不包含素数。

Explicit Short Intervals for Primes in Arithmetic Progressions on GRH

We prove explicit versions of Cramér's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

关于区间[3n，4n]中的素数

On the Primes in the Interval [3n, 4n]

对于区间[kn，（k+1）n]中是否总有素数这个老问题，著名的伯特兰公设对k=1给出了肯定的答案。这是由P.L.切比雪夫在1850年，和一个优雅的初等证明是由P.Erdos在1932年。M. El Bachraoui在2006年使用初等技巧证明了k=2的情况。本文给出了k=3情形的一个证明，同样没有使用素数定理或任何深层次的分析结果。此外，我们还给出了区间[3n，4n]中素数个数的一个下界，表明当n趋于无穷大时，区间[3n，4n]中素数个数趋于无穷大.

Prime polynomial values of quadratic functions in short intervals

In this paper we establish the function field analogue of Bateman-Horn conjecture in short interval in the limit of a large finite field. Hence we start with counting prime polynomials generated by primitive quadratic functions in short intervals. To this end we further work out on function field analogs of cancellation of Mobius sums and its correlations(Chowla type sums) and confirm that square root cancellation in Mobius sums is equivalent to square root cancellation in Chowla type sums.

关于含素数的区间（kn，（k+1）n）（n>1）

On intervals (kn,(k+1)n) containing a prime for all n>1

我们研究k的值，其中区间（kn，（k+1）n）包含一个素数，每个n>1。我们证明了这样的整数k的列表包括k= 1，2，3，5，9，14，没有其他的，至少对于k<= 50，000，000。对于这个列表的每个已知k，我们给出了最小N_k（m）的一个好的上估计，使得当n>=N_k（m）时，区间（kn，（k+1）n）至少包含m个素数。

连续素数之间大区间中某类加性函数的行为

The behaviour of a certain additive function in large intervals between consecutive primes

本文研究连续素数之间大区间内特定整数素因子所决定的加性函数的行为，结合素数大间隙相关结果、Erdős–Kac 定理及 Kubilius 模型得到主要结论。

Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} Λ(n) Λ(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost al…

关于短区间内素数定理的误差

On the error in the prime number theorem in short intervals

在黎曼假设下，用新平滑论证推导短区间素数定理误差项显式界，改进先前结果，并应用此界为短区间素数加权和建立更精确条件界。

Short intervals containing a prescribed number of primes

We prove that for every nonnegative integer $m$ there exists an $\varepsilon>0$ such that if $λ\in (0,\varepsilon]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$ for which the interval $[n,n+λ\log n]$ contains exactly $m$ primes is at least a constant times $x/\log x.$ This improves a result of T. Freiberg, when $λ$ is small.

短区间中的Piatetski-Shapiro素数

Piatetski-Shapiro Primes in short intervals

研究短区间中形如⌊n^c⌋的Piatetski-Shapiro素数的存在性，分别给出渐近公式和下界结果。

几乎所有短区间中的素数

Primes in almost all short intervals

本文改进了Jia的结果，证明在几乎所有长度为$n^{1/21.5+\varepsilon}$的短区间内必含素数，通过Watt均值界、筛法分解和精确积分估计实现。

Congruences with intervals and subgroups modulo a prime

We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),

短间隔的素数

Primes in short intervals

相反，将被预测的基础上Cramér的模型有关的素数的分布，我们开发的证据表明，$ψ(x+H)- ψ(x)$的分布，$0\le x\le N$，是近似正态的平均$\sim H$和方差$\sim H\log N/H$，当$N^δ\le H \le N^{1-δ}$。

素数在等差数列和短区间内的哥德巴赫问题

Goldbach's problem with primes in arithmetic progressions and in short intervals

本文讨论了几个类似于Bézieri-Vinogradov定理的中值定理。它们涉及在等差数列和短时间间隔内素数的二元和三元加法问题。其中一些平均值的非平凡估计。作为应用，我们证明了对于大奇数n \not \equiv 1（6），Goldbach三元问题n=p_1+p_2+p_3是可解的，其中素数p_1，p_2在短区间p_PDMATHABMO [X_i，X_i+Y]，且X_{i} ^{θ_i}=Y，i= 1，2，且θ_1，θ_2 \geq 0.933使得（p_1+2）（p_2+2）至多有9个素因子.

Short intervals asymptotic formulae for binary problems with prime powers, II

We improve some results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$ and $n=p^{\ell_1} + m^{\ell_2}$, where $\ell_1, \ell_2\ge 2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.

素数区间代数

Prime-Interval Algebras

研究从正整数\(n\)出发，在无先验信息时构造多项式商环，通过模指数运算恢复\((n,2n]\)内素数，仅用前一个素数就能得下一个素数，还发展了环结构、扩展结果并给出实现。

Almost primes in almost all very short intervals

We show that as soon as $h\to \infty$ with $X \to \infty$, almost all intervals $(x-h\log X, x]$ with $x \in (X/2, X]$ contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.

Sums of four prime cubes in short intervals

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones previously known.

短区间内无小素因子整数的方差

The variance of integers without small prime factors in short intervals

本研究聚焦短区间内无小素因子整数的方差问题，采用复分析与筛法理论得到某范围内的无条件渐近结果，发现该方差与y-光滑数统计相关，且区间长度达y的幂次时其渐近值小于朴素概率预测。

关于一定区间内素数个数的几个猜想

Some Conjectures on the Number of Primes in Certain Intervals

本文对素数作了一些比现有文献中更尖锐的分析。首先，我们描述了我们的研究勒让德的猜想，这仍然是未解决的。接下来，我们表明，布罗卡猜想可以证明假设我们的改进版本的勒让德猜想。最后，我们锐化了素数的Bertrand公设。我们的研究结果得到了广泛的实证研究的支持。

短区间内素数分布的偏差

Biases in the distribution of primes in short intervals

在Hardy-Littlewood素数元组假设下，研究短区间内素数计数分布的二级渐近，发现算术修正导致比Cramér模型更强的向均值偏差，并通过有限筛论证和数值比较验证。

短区间和小扇区中素数的Bombieri-Vinogradov定理

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。

A remark on the conditional estimate for the sum of a prime and a square

Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(x)$ be the number of positive integers up to $x\ge4$ which does not satisfy this condition. We prove $E(x)\ll x^{1/2}(\log x)^A(\log\log x)^4$with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement of the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claims $A=4,3$ respectively.

Primes in short intervals: Heuristics and calculations

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.