On large gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions
Johan Bredberg
本文证明了当$χ$是偶本原Dirichlet特征标时，某些Dirichlet $L$ -函数$L(s,χ),$在临界线上存在大的（平均值的3.54倍）连续零点间隙.
This text shows the existence of large (3.54 times the average) gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions $L(s,χ),$ with $χ$ being an even primitive Dirichlet character.
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Positive Proportion of Small Gaps Between Consecutive Primes
我们证明了一个积极的比例连续素数之间的差距是短的差距长度小于任何固定分数的平均间距之间的素数。
Primes in Tuples IV: Density of small gaps between consecutive primes
我们表明，一个积极的比例连续素数之间的所有差距是小差距。我们提供了几个定量的结果，一些无条件和一些有条件的，在这种味道。
Bounded gaps and perfect power gaps in sequences of consecutive primes
本文研究连续素数间隙中是否存在较大间隙或完美幂间隙，并探讨相关数学问题。
On the ratio of consecutive gaps between primes
在目前的工作中，我们证明了一个共同的推广梅纳德-陶最近的结果连续有界的差距之间的素数和Erdens-Rankin界之间的连续素数的大差距。这项工作的答案在一个强有力的形式60岁的问题埃尔德什，问是否比两个连续primegaps可以无限经常任意小，和任意大，分别。
Large gaps between consecutive zeros of the Riemann zeta-function. II
假设黎曼假设，我们表明，存在无穷多个连续的零点的黎曼ζ函数的间隙是大于2.9倍的平均间距。
Large gaps between consecutive prime numbers containing perfect powers
对于任何正整数$k$，我们表明，无限经常，完美的$k$ -th权力出现在非常长的间隙内连续素数之间，即间隙的大小$$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$其中$p$是较小的两个素数。
Large gaps between consecutive zeros of the Riemann zeta-function
结合的软化，我们表现出其他的选择系数，改善结果之间的大差距的黎曼ζ函数的零点。准确地说，假设广义黎曼假设（GRH），我们表明，存在无穷多个连续的间隙大于大于3.033倍的平均间距。
We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$π(x)$ --- the number of primes up to: $M_k(x)= Γ(k+1)x^k/π^{k-1}(x)+\mathcal{O}(x)$, We illustrate obtained results by computer data.
Large gaps between consecutive prime numbers containing square-free numbers and perfect powers of prime numbers
我们证明了K. Ford，D. R. Heath-Brown和S. Konyagin关于无平方数的素数回避和素数的完全幂。
A note on the gaps between consecutive zeros of the Riemann zeta-function
假设的黎曼假设，我们表明，无限经常连续的非平凡零的黎曼ζ函数相差最多0.5155倍的平均间距和无限经常他们相差至少2.69倍的平均间距。
Small gaps between consecutive zeros of the Riemann zeta-function
本文引入共振相关方法研究黎曼ζ函数连续零点间的小间隔，结合Montgomery对称方法和Montgomery-Odlyzko方法，突破0.515障碍，证明在黎曼假设下μ<0.50895。
We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $π(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{π^2(x)}{x-2π(x)}\log\Big(\frac{x}{2π(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.
A small improvement in the small gaps between consecutive zeros of the Riemann zeta-function
Feng和Wu在Montgomery和Odlyzko的方法中引入了一个新的通用系数序列，用于在假设黎曼假设的情况下，显示$ζ(s)$的连续零点之间的间隙的不规则性。他们使用了一个特殊的情况下，他们的序列，以改善早期的结果的差距。本文考虑Feng和Wu的一般序列的一个等价形式，并引入了一个不太一般的序列$\{a_n\}$，我们明确地写出了它的Montgomery-Odlyzko表达式.作为应用，我们给出了Feng和Wu的结果的如下改进：黎曼ζ函数的无穷次连续非平凡零点至多相差平均间距的$0{.}515396$倍.
Large gaps between consecutive maxima of the Riemann zeta-function on the critical line
本文在满足黎曼假设的前提下，得到了临界线上黎曼ζ函数的相邻极大值之间的归一化距离的新下界.我们的证明方法依赖于一个一维Sobolev型不等式和一个具有最佳常数的Opial型不等式。
Large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function
我们证明了对于任何足够大的$T,$，存在一个长度至少为$2.766 \times \frac{2π}{\log{T}},$的$[T,2T]$的子区间，其中函数$t \mapsto ζ(1/2 + it)$没有零。
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 propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.
On a conjecture of ErdŐs, Pólya and Turán on consecutive gaps between primes
设p_n表示所有素数的序列，d_n=p_n- p_{n-1}表示相邻素数之间所有间隔的序列.在1948年Erdés和Turán表明，d_{n+1} -d_n的变化标志无限经常，并连同Pólya要求的一个充要条件，一个固定的线性组合r>1连续的总理差距d_{n+i} i=（1，2，. r）应该随着n贯穿所有自然数的序列而无限频繁地改变符号。他们指出，条件是线性组合的系数之间的非零值不能都具有相同的符号，但他们只能证明弱得多的关系。本文借助于Maynard-Tao方法和W.D.班克斯，T。Freiberg和J.Maynard（参见arXiv：1404.59094v2）。
Consecutive moderate gaps between zeros of the Riemann zeta function
本文研究了Riemann zeta函数非平凡零点纵坐标间的间隙问题，探讨了在给定条件下是否无限次存在连续的“中等”间隙。
Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many values of $n$. As it is shown, the key ingredient to obtain this behaviour is a consequence of an extension of the Kummer's characterization of convergent series of positive terms.
Large gaps between consecutive prime numbers
令$G(X)$表示低于$X$的连续素数之间的最大间隙的大小。讨论了Erdos的一个问题，证明了$$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$，其中$f(X)$是一个趋于无穷大的函数，且具有$X$ .我们的证明结合了现有的参数与随机建设覆盖一组素数的等差数列。因此，我们依赖于最近的工作的存在性和分布的长等差数列完全由素数。
Expected gaps between prime numbers
我们直接通过Eratosthenes筛法来研究连续素数之间的间隔。使用初等方法，我们确定了这些差距和特定序列的连续差距，称为星座的递归关系。使用这种递归，我们可以估计一个素数和它的平方之间出现的间隙或星座的数量。这种递归也有明确的含义开放的问题之间的差距素数，包括三个问题提出的埃尔德什和图兰。
Jumping champions and gaps between consecutive primes
小于或等于$x$的连续素数之间最常见的差异称为跳跃冠军。偶尔也会有联系。因此，对于给定的$x$，可以有多个跳跃冠军。1999年A.奥德利兹科湾Rubinstein和M.沃尔夫提供了启发式和经验证据来支持这样一个猜想，即大于大于1的跳跃冠军是4，素数是2、6、30、210、2310.作为证明这一猜想的一步，他们引入了第二个较弱的猜想，即任何固定的素数$p$都可以整除所有足够大的跳跃冠军。本文推广了P.Erdens和E. G. Straus于1980年提出了第二个猜想，证明了第二个猜想是由G. H. Hardy和J.E.利特尔伍德
The behaviour of a certain additive function in large intervals between consecutive primes
本文研究连续素数之间大区间内特定整数素因子所决定的加性函数的行为，结合素数大间隙相关结果、Erdős–Kac 定理及 Kubilius 模型得到主要结论。
On moments of gaps between consecutive squarefree numbers
本文研究了连续无平方因子数之间间隙的矩，证明了相关渐近式在 $0 \le γ< 3.75$ 范围内成立，将此前 $0 \le γ< 3.6875$ 的最佳范围予以改进。
We show that large gaps between smooth numbers are infrequent. The key new tool is a novel mean value bound for a special type of Dirichlet polynomial.
Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes…
Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers
在以前的论文中，作者证明了在任何满足明显必要局部条件的三个线性型的系统中，至少有两个形式无限地经常假设$E_2$ -值;即，两个素数的乘积。我们利用这个结果证明了有无穷多个整数$x$同时满足$$ω(x)=ω(x+1)=4, Ω(x)=Ω(x+1)=5, \text{and} d(x)=d(x+1)=24.$$这里，$ω(x), Ω(x), d(x)$分别表示$x$的素除子数、$x$的素幂因子数和$x$的因子数。对于任意正整数$b$，我们也证明了类似的定理，其中$x+1$被$x+b$取代。我们的结果加强了Heath-Brown，Pinner和Schlage-Puchta的早期工作。
Gaps Between Consecutive Primes and the Exponential Distribution
本文基于小素数间隔数据，证明 Oliveira e Silva 的幂和猜想等价于间隔矩的指数分布渐近性质，并由此联系 Taylor 定律、最大间隔期望与 Cramér-Shanks 猜想，数值结果还提示相关 limsup 可能超过 1。
On the distribution of gaps between consecutive primes
Erdös证明了d_n/logn的极限点的集合J包含所有非负数，其中d_n表示第n个素隙。作者在一年前（arXiv：1305.6289）证明了J包含一个[0，c]型区间，其无效值c为正。在目前的工作中，我们扩展了这一结果的一大类正规化功能。唯一的基本要求是，替换logn的函数f（n）应满足f（n）<<lognloglognloglogloglogloglogn/（logloglogn）^2（有一个小的隐含常数），即著名的连续素数之间最大已知间隔的埃尔德什-兰金界。工作还证明了，除了一个薄集的例外功能，原来的埃尔德什猜想成立，如果logn被替换为一个非例外功能f（n）。本文还对Helmut Maier的一个结果给出了新的证明，该结果推广了任意长的有限连续素隙链的Erdös-Rankin界.证明使用了Erdös-Rankin，Maynard-Tao和Banks-Freiberg-Maynard的方法的组合。自从本论文发表以来，福特-格林-魔芋-陶（arXiv：1408.4505 [math.NT]）和梅纳德（aerXiv：1408.5110 [math.NT]）的非常重要的近期同步和独立的工作出现在arXiv上，他们证明了埃尔德什的旧猜想，该猜想断言大间隙的下界超过了具有任意大常数C的Clognloglognloglogloglogn/（logloglogn）^2。在这个新版本中，我们…
Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum
研究一维Dirichlet谱中射线起点μ的上下界改进，证明Ivanov点Z_k为孤立点，并刻画其邻域间隙渐近行为及等价类。
Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture
在本工作中，我们证明了一些令人惊讶的结果之间的差距连续素数和等差数列的广义孪生素数序列，不可能被证明没有最近的神奇成就张一堂连续素数之间存在有界的差距。这些结果中的大多数在十年前都属于科幻小说的范畴。然而，所提出的结果是远远没有直接后果张的著名定理：他们需要各种新的想法，其他重要的性质的应用筛功能和更密切的分析方法的Goldstone-Pintz-Yildirim，绿色陶，和张，分别。
Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$ for some small constant $c_m>0$. Furthermore, we also obtain a related result concerning the least primes in arithmetic progressions.
On the constant in Burgess' bound for the number of consecutive residues or non-residues
我们给出了D的一个结果的一个显式版本。伯吉斯设$χ$是模素数$p$的非主狄利克雷特征标。我们表明，$χ$采取一个特定的值的连续整数的最大数量小于$\left\{\frac{πe\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$，其中的$o(1)$长期明确给出。
On large gaps between zeros of the Riemann zeta-function
假设广义黎曼假设（GRH），我们证明了黎曼ζ函数的无限连续非平凡零点的差异至少是平均间距的3.072倍。
Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.
We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.
Small gaps between zeros of twisted L-functions
本文利用作者提出的渐近大筛证明了：如果广义黎曼假设成立，则存在许多Dirichlet L函数的一对连续零点的距离小于其平均间距的0.37倍.更一般地说，我们调查零间距家庭内的扭曲的Dirichlet字符的一个固定的L-函数，并给出精确的界限小的差距，只依赖于程度的L-函数。
Small Gaps Between Primes I
我们使用短除数和来近似素数元组和素数在短间隔的时刻。通过将这些结果与经典矩问题相联系，我们能够证明连续素数的正比例在素数之间的平均间距的四分之一以内。
We study the value-distribution of Dirichlet polynomials on the critical line $\Re(s)=\tfrac{1}{2}$. As a consequence, we prove a corollary on small consecutive gaps between zeros of the Riemann zeta function. We also examine the distribution of zeros under the so-called alternative hypothesis and present a new approach to the problem of gaps between the zeros.
We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).
On gaps between zeros of the Riemann zeta function
假设的黎曼假设，我们表明，无限经常连续的非平凡零点的黎曼ζ函数相差至少2.7327倍的平均间距和无限经常他们相差最多0.5154倍的平均间距。
On the difference between consecutive primes
更新：这项工作再现了Peck的早期结果，作者最初并不知道。证明的方法与Peck的原始工作基本相同。没有新的结果。我们证明了连续素数$\sum_{p_n\le x}(p_{n+1}-p_n)^2$之间的差的平方和是有界的$x^{5/4+ε}$的$x$足够大，任何固定的$ε>0$。证明依赖于利用各种均值估计狄利克雷多项式。
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper …
In this paper, we apply the method of Maynard and Tao to the set of products of two distinct primes (E2-numbers). We obtain several results on the distribution of E2-numbers and primes. Among others, the result of Goldston, Pintz, Yildirim and Graham on small gaps between m consecutive E2-numbers is improved.
Let $K$ be a quadratic number field and $ζ_K(s)$ be the associated Dedekind zeta-function. We show that there are infinitely many normalized gaps between consecutive zeros of $ζ_K(s)$ on the critical line which are greater than $2.866$ times the average spacing.
Bounded gaps between primes with a given primitive root, II
令$m$为自然数，令$\mathcal{Q}$为至少包含$\exp(C m)$素数的集合。我们发现，可以找到无限多个字符串的$m$连续素数，其中每个有一些$q\in\mathcal{Q}$作为一个原根，所有位于一个区间的长度$O_{\mathcal{Q}}(\exp(C'm))$。这是古普塔和拉姆·默蒂定理的一个有界间隙变体。我们还证明了一个结果的椭圆模拟阿廷猜想。设$E/\mathbb{Q}$是一条椭圆曲线，它有一个无理$2$ -扭点.假设GRH。则对于每个$m$，有无穷多个$m$连续素数$p$的字符串，其中$E(\mathbb{F}_p)$是循环的，所有字符串都位于长度为$O_E(\exp(C'' m))$的区间内。如果$E$具有CM，则可以移除GRH假设。这里$C$、$C'$和$C''$是绝对常数。
On Maximal Prime Gaps
本文研究了素数间隙的问题，证明了素数与其后继素数之间的间隙不超过2log²p_n，并由此推导出对于足够大的数，存在特定类型的区间内必然存在素数。
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.
Large gaps between the zeros of the Riemann zeta function
证明了广义黎曼假设意味着黎曼ζ函数存在无穷多个连续零点，其零点间距是平均零点间距的2.9125倍.这是从计算黎曼ζ函数的二阶矩乘以在ζ函数的零点上平均的狄利克雷多项式推导出来的。
On small values of the Riemann zeta-function on the critical line and gaps between zeros
利用A.塞尔伯格这给出了$μ(\{0 < t \le T : |ζ(1/2+it)| \le c\})$的渐近公式。本文还讨论了有关$|ζ(1/2+it)|$的值和$ζ(s)$的零点的连续纵坐标之间的间隙的一些问题。
A note on bounded gaps between primes
作为张义堂最近工作的一个改进，我们证明了：当k至少为181000时，任意可容许的k-整数元组在每一个组成部分中无限频繁地包含至少两个素数和几乎素数.这意味着在连续素数之间存在无穷多个间隔，其大小至多为2530338，并且Andrew W.萨瑟兰呼叫2326476
On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes
我们解决了最优化问题. Soundararajan，用于选择在Goldston-Pintz-Yildirim方法中用于构造权函数的基本多项式P，以寻找素数之间的小间隙。首先，我们将L ^2 [0，1]上的极大化问题转化为一个自伴算子T，其范数是T的最大特征值。为了找到本征函数和本征值，我们导出了一个可以显式求解的微分方程。目标最大值是S（k）=4/（k+ck^{1/3}），通过x^{1-k/2} J_{k-2}（a_1 \sqrt{x}）的第（k-1）次积分实现，其中a_1是第（k-2）次贝塞尔函数J_{k-2}的第一个正根，因此，渐近地为ck^{1/3}，具有众所周知的常数c。由于这自然会在GPY方法的应用中产生许多技术问题，我们还构造了一个多项式P，它是一个更简单的函数，但它用其他常数C来逼近近似最佳极值量4/（k+Ck^{1/3}）。在即将发表的J. Pintz的论文中（也将出现在de Gruyter的Turán-100 Memorial Volume中），它确实表明了这种准最优选择的多项式的权重最终可以利用GPY方法达到其理论极限。
On the frequency of small gaps between the primes
本文针对连续素数间隙倒数和发散所需间隙大小的问题，在Friedlander依赖深层k元组猜想的结果之外，对其弱化形式给出了无条件解答。
An explicit lower bound for large gaps between some consecutive primes
本文确定了相邻素数间大间隙下界不等式中的有效绝对常数 $c_{LG}$，证明其至少可取值为 $2.0\times 10^{-17}$，该常数受 Maynard 多维筛法、算术级数素数分布及 Selberg 上界筛法系数等因素决定。
Rough numbers between consecutive primes
本文用筛法证明相邻素数间隙中几乎总存在最小素因子不小于间隙长度的数，并给出例外间隙数的上界及在猜想下的渐近式。
Gaps between zeros of GL(2) $L$-functions
设$L(s,f)$是与$\mathbb{Q}$上GL（2）上的本原（全纯或Maass）尖点形式$f$相关联的$L$ -函数.结合蒙哥马利和沃恩的平均值估计与Ramachandra的方法，我们证明了一个公式的混合二阶矩的衍生物的$L(1/2+it,f)$和，通过霍尔的方法，使用它来显示，有无穷多个间隙之间的连续零的$L(s,f)$沿临界线，至少是平均间距的$\sqrt 3 = 1.732...$倍。使用一般对相关的结果，由于Murty和Perelli结合蒙哥马利的技术，我们还证明了存在的零之间的任何原始$L$ -函数的Selberg类的小差距。特别地，当$f$是$\mathbb{Q}$上GL（2）上的本原全纯尖点形式时，我们证明了$L(s,f)$的连续零点之间沿临界线存在无穷多个间隙，这些间隙至多是平均间距的$< 0.823$倍.
If $a$ and $d$ are relatively prime, we refer to the set of integers congruent to $a$ mod $d$ as an `eligible' arithmetic progression. A theorem of Dirichlet says that every eligible arithmetic progression contains infinitely many primes; the theorem follows from the assertion that every eligible arithmetic progression contains at least one prime. The Jacobsthal function $g(n)$ is defined as the smallest positive integer such that every sequence of $g(n)$ consecutive integers contains an integer relatively prime to $n$. In this paper, we show by a combinatorial argument that every eligible ari…
In 1931, Van der Corput showed that if for each positive integer $s$, the sequence $\{x_{n+s}-x_n\}$ is uniformly distributed (mod 1), then the sequence $x_n$ is uniformly distributed (mod 1). The converse of above result is surprisingly not true. The distribution of consecutive gaps of an equidistributed sequence has been studied widely in the literature. In this paper, we have studied the distribution of gaps between one or more equidistributed sequences. Under certain conditions, we could study the distribution effectively. As applications, we study the equidistribution of gaps between eige…
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