Primes in explicit short intervals on RH

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

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

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.

Very small intervals containing at least three primes

Let $p_n$ is the $n$-th prime. With help of the Cramér-like model, we prove that the set of intervals of the form $(2p_n,\enskip2p_{n+1})$ containing at list 3 primes has a positive density with respect to the set of all intervals of such form.

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

Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.

Higher moments of primes in short intervals I

In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.

Bounded length intervals containing two primes and an almost-prime

Goldston, Pintz and Yıldırım have shown that if the primes have `level of distribution' $θ$ for some $θ>1/2$ then there exists a constant $C(θ)$, such that there are infinitely many integers $n$ for which the interval $[n,n+C(θ)]$ contains two primes. We show under the same assumption that for any integer $k\ge 1$ there exists constants $D(θ,k)$ and $r(θ,k)$, such that there are infinitely many integers $n$ for which the interval $[n,n+D(θ,k)]$ contains two primes and $k$ almost-primes, with all of the almost-primes having at most $r(θ,k)$ prime factors. If $θ$ can be taken as large as $1-ε$, …

Higher moments of primes in short intervals II

Given good knowledge on the even moments, we derive asymptotic formulas for $λ$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.

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.

On critical small intervals containing primes

Let p be an odd prime, such that p_n<p/2<p_{n+1}, where p_n is the n-th prime. We study the following question: with what probability does there exist a prime in the interval (p, 2p_{n+1})? After the strong definition of the probability with help of the Ramanujan primes ([11], [12])and the introducing pseudo-Ramanujan primes, we show, that if such probability P exists, then P>=0.5. We also study a symmetrical case of the left intervals, which connected with sequence A080359 at OEIS.

Sums of primes and squares of primes in short intervals

Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.

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.

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.

Bounded prime gaps in short intervals

We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x (\log x)^{-A}]$. Our result follows only by analysing Zhang's proof of Theorem 1, but we also explain how a sharper variant of Zhang's Theorem 2 would imply the same result for shorter intervals.

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

We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent equidistribution results of N. Katz.

More precise pair correlation of zeros and primes in short intervals

Goldston and Montgomery [3] proved that the Strong Pair Correlation Conjecture and two second moments of primes in short intervals are equivalent to each other under Riemann Hypothesis. In this paper, we get the second main terms for each of the above and show that they are almost equivalent to each other.

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 $n>5$.

A note on Primes in Short Intervals

Instead of a strong quantitative form of the Hardy-Littlewood prime $k$-tuple conjecture, one can assume an average form of it and still obtains the same distribution result on $ψ(x+h) - ψ(x)$ by Montgomery and Soundararajan [1].

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.

Note On Prime Gaps And Very Short Intervals

Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <= c1((logx)^2)/loglogx, c1 > 0 constant. Equivalently, it shows that the very short intervals (x, x + y] contain prime numbers for all y > c2((logx)^2)/loglogx, c2 > 0 constant, and sufficiently large x > 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.

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。

Prime numbers in logarithmic intervals

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every $λ>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ λ\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $λ>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ λ\log X]$ contains no prime…

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.

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

For the old question whether there is always a prime in the interval [kn, (k+1)n] or not, the famous Bertrand's postulate gave an affirmative answer for k=1. It was first proved by P.L. Chebyshev in 1850, and an elegant elementary proof was given by P. Erdos in 1932. M. El Bachraoui used elementary techniques to prove the case k=2 in 2006. This paper gives a proof of the case k=3, again without using the prime number theorem or any deep analytic result. In addition we give a lower bound for the number of primes in the interval [3n, 4n], which shows that as n tends to infinity, the number of pr…

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.

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

We study values of k for which the interval (kn,(k+1)n) contains a prime for every n>1. We prove that the list of such integers k includes k=1,2,3,5,9,14, and no others, at least for k<=50,000,000. For every known k of this list, we give a good upper estimate of the smallest N_k(m), such that, if n>=N_k(m), then the interval (kn,(k+1)n) contains at least m primes.

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),

Primes in short intervals

Contrary to what would be predicted on the basis of Cramér's model concerning the distribution of prime numbers, we develop evidence that the distribution of $ψ(x+H)- ψ(x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^δ\le H \le N^{1-δ}$.

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

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.

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。

Some Conjectures on the Number of Primes in Certain Intervals

In this paper, we make some conjectures on prime numbers that are sharper than those found in the current literature. First we describe our studies on Legendre's Conjecture which is still unsolved. Next, we show that Brocard's Conjecture can be proved assuming our improved version of Legendre's Conjecture. Finally, we sharpen the Bertrand's Postulate for prime numbers. Our results are backed by extensive empirical investigation.

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。

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.

On sum of Hecke eigenvalue squares over primes in very short intervals

Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.

拟模形式、分拆与区间中的素数

Quasimodular Forms, Partitions, and Primes in Intervals

本文利用素数探测拟模形式的性质，证明素数无穷性，构造基于分拆的因式分解算法，并揭示系数分布与区间素数存在性的关联，同时推广至算术级数及Linnik问题。

数域中短区间素数加权矩的下界

Lower bounds on weighted moments of primes in short intervals in number fields

本研究针对数域中短区间素数矩的Montgomery与Soundararajan猜想类似问题，通过调整整数情形下的已有研究方法，建立了加权素数矩的下界，且结果与猜想值相符。