Tsz Ho Chan
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].
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Zhang has shown there are infinitely many intervals of bounded length containing two primes. It appears that the current techniques cannot prove that there are infinitely many intervals of bounded length containing three primes, even if strong conjectures such as the Elliott-Halberstam conjecture are assumed. We show that there are infinitely many intervals of length at most $10^8$ which contain two primes and a number with at most 31 prime factors.
The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
We prove the analog of Cramér's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.
Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.
Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any given length.
Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+ε}p_n)$ where $ε> 0$, in the context of the results we have obtained in our paper.
We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.
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.
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.
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.
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.
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 $1-ε$, …
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.
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…
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.
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.
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.
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.
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.
On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.
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.
We survey the classical results on the prime number theorem
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.
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.
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.
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…
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.
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 $n>5$.
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.
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.
By combining a sieve method of Harman with the work of Maynard and Tao we show that $$\liminf_{n\rightarrow \infty}(p_{n+m}-p_n)\ll \exp(3.815m).$$
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 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.
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.
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。
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…
We prove explicit versions of Cramér's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.
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…
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.
We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.
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.
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。
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…
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。
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.
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。
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。
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),
We investigate the distribution of $pθ$ modulo 1, where $θ$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.
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 $N^δ\le H \le N^{1-δ}$.
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.
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.
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。
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.
Let $α, β\geq 0$ and $α+ β< 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of Sándor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.
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.
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。
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.
In this note, we revisit a result of Benli's and Pollack's on the number of small prime $k^{th}$ power residues. The proof is based on their idea of using reciprocity laws, but the argument is simplified and we prove a slightly stronger bound.
Let p be a prime number, and let K be a number field. For p=2, assume moreover K totally imaginary. In this note we prove the existence of asymptotically good extensions L{K of cohomological dimension 2 in which infinitely many primes split completely. Our result is inspired by a recent work of Hajir, Maire, and Ramakrishna [7].
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