Anton Deitmar
The bounded gaps property of the prime numbers, as proven by Yitang Zhang, is considered for sequences of lengths of closed geodesics, which by the theory of Selberg zeta functions are the geometric analogue of the prime numbers. It turns out that the property holds for congruence subgroups and is false for a dense set in Teichmüller space.
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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。
Let $r \ge 2$ be an integer. We adapt the Maynard-Tao sieve to produce the asymptotically best-known bounded gaps between products of $r$ distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.
Bounded gaps between primes in special sequences
利用Maynard方法证明了序列$\{\lfloor nα\rfloor\}$中素数之间存在有界间隙，其中$α$是有限型的无理数.另外，给出了一个满足Leitmann性质的超线性函数$f$，证明了对于所有的$m$，存在无穷多个包含$m$素数和至少一个$\lfloor f(q)\rfloor$形式的整数的有界区间，其中$q$是正整数.
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。
Using Duke's large sieve inequality for Hecke Gr{ö}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ε<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<ε\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:ax^α+by^β=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<ε\sqrt{p}$.
For each $m\geq 1$, there exist infinitely many primes $p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
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
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。
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。
Bounded gaps between primes in number fields and function fields
Hardy-Littlewood素数$k$ -元组猜想一直被认为是用现有方法完全不可接近的。虽然这仍然是事实，但张，梅纳德和陶的惊人突破仍然在这个问题上取得了重大进展。本文将Maynard-Tao方法推广到数域和函数域$\mathbb{F}_q(t)$。
A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.
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…
Winning games for bounded geodesics in moduli spaces of quadratic differentials
我们证明了Teichmuller空间中的有界测地线集合是Schmidt博弈的获胜集合。这是度量空间中的一个大概念，可以应用于测度0和稀疏集。我们证明类似的密切相关的结果在任何黎曼曲面，在任何阶层的二次微分，在任何Teichmuller盘和任何固定的不可约置换的间隔交换。
We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston-Graham-Pintz-Yildirim \cite{GGPY}, and Maynard \cite{MAY}. An important consequence of our main theorem is existence of infinitely many pairs $α_1, α_2$ which are product of two primes in the imaginary quadratic field $K$ such that $|σ(α_1-α_2)|\leq 2$ for all embedding $σ$ of $K$ if the class number of $K$ is one and $|σ(α_1-α_2)|\leq 8$ for all embedding $σ$ of $K$ if the class number of $K$ is two.
On small gaps between primes and almost prime powers
在最近与D.A. Goldston和C.Y. Yildirim我们刚刚错过了一个证明素数之间有界间隙无限频繁出现的证据。在目前的工作表明，添加到素数一个更薄的一套，所谓的几乎素数权力，工会的一套素数和几乎素数权力已经包含无限多有界的差距。更确切地说，它表明，如果我们添加到一组素数要么几乎素平方正好有两个，几乎相等的素因子，或者如果我们添加到一组素数几乎素立方体正好有三个，几乎相等的素因子，那么所产生的集合已经包含无限多个有界间隙。
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic set…
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。
Bounded gaps between primes
这些笔记是张的工作和随后的发展产生的准备5个小时的会谈一般数学观众在剑桥，爱丁堡和奥克兰在过去的一年。由于是学术讨论会式的谈话，这些笔记比文献中的其他记录要低得多。在某些地方，他们故意相当不严谨。是否有人会发现它们有帮助还不清楚，但在准备它们时已经采取了一些措施，我希望在这里提供它们没有什么害处。
Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory
本文提出融合混沌扰动与RMT的EMCHS概率筛框架，在GPY与Maynard筛法基础上启发式改进素数间隙界，并用数值证据与部分严格证明加以支撑。
Bounded gaps between primes with a given primitive root
修复一个不是完全平方的整数$g \neq -1$。在1927年，阿廷证明有无穷多个素数$g$是一个原根。40年后，胡利证明了阿廷猜想来自广义黎曼假设（GRH）。我们注入胡利的分析梅纳德-陶工作有界素数之间的差距。这将导致以下GRH条件结果：固定一个整数$m \geq 2$。如果$q_1 < q_2 < q_3 < \dots$是以$g$为原根的素数序列，则为$\liminf_{n\to\infty} (q_{n+(m-1)}-q_n) \leq C_m$，其中$C_m$是依赖于$m$但不依赖于$g$的有限常数。我们还证明了这个结果中的素数$q_n, q_{n+1}, \dots, q_{n+m-1}$可以看作是连续的。
The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms of the Hausdorff-dimension of suitable subsets of the set of bounded geodesic rays. We establish estimates on the Hausdorff-dimension of these subsets and thereby obtain non-trivial bounds for the dimension functions. Moreover we discuss the property of B being an absolute winning set, therefore satisfying a remarkable rigidity. Finally, we apply the obtaine…
The remainder $E_Γ(X)$ in the Prime Geodesic Theorem for the Picard group $Γ= \mathrm{PSL}(2,\mathbb{Z}[i])$ is known to be bounded by $O(X^{3/2+ε})$ under the assumption of the Lindelöf hypothesis for quadratic Dirichlet $L$-functions over Gaussian integers. By studying the second moment of $E_Γ(X)$, we show that on average the same bound holds unconditionally.
On primes in arithmetic progressions and bounded gaps between many primes
本研究改进了算术级数中素数分布的指数，修正q-van der Corput过程以利用指数和的额外平均，进而优化了多素数有界间隙的相关结果。
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。
Small gaps between Goldbach primes
本文通过Bombieri-Davenport方法和Maynard-Tao方法研究Goldbach素数间的极小间隙，改进了Tsuda的结果并证明了Goldbach素数间存在有界间隙。
Let $(X,B_X,μ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ with $μ(A)>0$ and $ε>0$. For any $m\geq 1$, there exist infinitely many primes $p_0,p_1,\ldots,p_m$ with $p_0<\cdots<p_m$ such that $$ μ(A\cap T^{-(p_i-1)}A)\geq μ(A)^2-ε$$ for each $0\leq i\leq m$ and $$ p_m-p_0<C_m, $$ where $C_m>0$ is a constant only depending on $m$, $A$ and $ε$.
On the law of the iterated logarithm for trigonometric series with bounded gaps II
众所周知，对于快速增加的序列$(n_k)_{k \geq 1}$，函数$(\cos 2 πn_k x)_{k \geq 1}$表现出独立随机变量序列的典型行为。如果放松$(n_k)_{k \geq 1}$上的生长条件，则这种几乎独立的行为通常会失败。尽管如此，概率构造表明，对于\emph{some}非常缓慢地增加的序列$(n_k)_{k \geq 1}$，这种几乎独立的属性被保留。例如，存在具有有界间隙的$(n_k)_{k \geq 1}$，使得归一化和$\sum \cos 2 πn_k x$满足中心极限定理（CLT）。然而，由于“质量损失”现象，具有有界空位的序列的CLT的方差总是小于$1/2$。在这种情况下的法律的重对数（LIL）的情况是不同的，因为我们证明了在较早的文件中，存在$(n_k)_{k \geq 1}$有界的差距，使$$ \limsup_{N \to \infty} \frac{\left| \sum_{k=1}^N \cos 2 πn_k x \right|}{\sqrt{N \log \log N}} = \infty \qquad \textrm{for almost all $x$.} $$在本文中，我们证明了一个互补的结果表明，任何规定的limsup行为的LIL序列有界的差距是可能的。更准确地说，我们证明了，对于任何实数$Λ\geq 0$存在一个整数序列$…
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