John Friedlander, Henryk Iwaniec
We study certain aspects of the Selberg sieve, in particular when sifting by rather thin sets of primes. We derive new results for the lower bound sieve suited especially for this setup and we apply them in particular to give a new sieve-propelled proof of Linnik's theorem on the least prime in an arithmetic progression in the case of the presence of exceptional zeros.
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We give a simple proof of a standard zero-free region in the $t$-aspect for the Rankin--Selberg $L$-function $L(s,π\times \widetildeπ)$ for any unitary cuspidal automorphic representation $π$ of $\mathrm{GL}_n(\mathbb{A}_F)$ that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.
We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry integral" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the "classical" level, up to level of distribution, say, lambda=log Q/log N < 2/3 (for enough large h). This time we don't apply Large Sieve inequality, as in our paper [C-S]. Precisely, our method is completely elementary.
Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
本文证明对每个符号，存在大量（\\(\gg X/\log X\\)）至多四个素因子的无平方因子模\\(q\\)，其归一化Kloosterman和取该符号，方法结合Selberg筛与等分布及均方估计。
Let $k\geq 2$ and $\mathcal{P} (n) = (A_1 n + B_1 ) \cdots (A_k n + B_k)$ where all the $A_i, B_i$ are integers. Suppose that $\mathcal{P} (n)$ has no fixed prime divisors. For each choice of $k$ it is known that there exists an integer $\varrho_k$ such that $\mathcal{P} (n)$ has at most $\varrho_k$ prime factors infinitely often. We used a new weighted sieve set-up combined with a device called an $\varepsilon$-trick to improve the possible values of $\varrho_k$ for $k\geq 7$. As a by-product of our approach, we improve the conditional possible values of $\varrho_k$ for $k\geq 4$, assuming th…
A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z). Consequently we obtain sharp upper bounds in the affine linear sieve, where in contrast to \cite{BGS} we use an archimedean norm to order the elements.
An arithmetic function $f$ is called a sieve function of range $Q$, if it is the convolution product of the constantly $1$ function and $g$ such that $g(q)\ll_{\varepsilon} q^{\varepsilon}$, $\forall\varepsilon>0$, for $q\leq Q$, and $g(q)=0$ for $q>Q$. Here we establish a new result on the autocorrelation of $f$ by using a famous theorem on bilinear forms of Kloosterman fractions by Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain non-trivial asymptotic formulæ that are actually unreachable by the standard approach of the distribution of $f$ in the arithmetic prog…
Non-orthogonality of the cubic and quartic large sieves via Rankin-Selberg
研究三次和四次大筛法的非正交性，利用久保田狄利克雷级数二阶矩平均林德勒夫上界及亚纯θ函数兰金 - 塞尔伯格卷积傅里叶系数平均下界，采用扎吉尔正则化方法，还对特定赫克特征族提出猜想。
Selberg sieve weights and sign changes of Kloosterman sums. I. Uniform asymptotics for shifted weights
本文证明带二次截断函数的除数和关联的均匀渐近公式，误差为O_g(Y/(log Y)^2)，主项为导数双线性形式，为Kloosterman和符号变化问题提供关键估计。
For any $m \geq 1$, let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit bound $H_1 \leq 70000000$. This was then improved by us (the Polymath8 project) to $H_1 \leq 4680$, and then by Maynard to $H_1 \leq 600$, who also established for the first time a finiteness result for $H_m$ for $m \geq 2$, and specifically that $H_m \ll m^3 e^{4m}$. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound $H_1 \leq 12$, improving upon the previous bound $H_1 \leq 16$ of Goldsto…
We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg sieve.
The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a_1,...,a_k and b_1,...,b_k be positive integers. For t on the unit circle write h(t) := \sum_{n \in X} e(nt)$, where X is the set of all n <= N such that the numbers a_1n + b_1,..., a_kn + b_k are all prime. We obtain upper bounds for the L^p norm of h, p > 2, which are (conditionally on the prime tuple conjecture) of the corre…
Sifting limits for the $Λ^{2}Λ^{-}$ sieve, Selberg's lower bound sieve, are computed for integral dimensions $1<κ\le10$. The evidence strongly suggests that for all $κ\ge3$ the $Λ^{2}Λ^{-}$ sieve is superior to the competing combinatorial sieves of Diamond, Halberstam, and Richert. A method initiated by Grupp and Richert for computing sieve functions for integral $κ$ is also outlined.
We study an instance of the Affine Sieve, producing a level of distribution beyond that which can be obtained from current techniques, even assuming a Selberg/Ramanujan-type spectral gap. In particular, we consider the set of hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07, Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in this linear sieve problem (see Sec. 1 for definitions). If conditioned on a best possible gap theta = 1/2, the known method would give an exponent alpha…
A note on $θ_2$
本文针对素数模数用光滑权无条件重证 Selberg 的除数函数分布水平 $\theta_2=2/3$，并提出借助次凸性提升至 $4/5$ 的假设，以期改进孪生素数逼近和素数间隔下界。
We show how the Selberg $Λ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.
Selberg identified the "parity" barrior, that sieves alone cannot distinguish between integers having an even or odd number of factors. We give here a short and self-contained demonstration of parity breaking using bilinear forms, modeled on the Twin Primes Conjecture.
Large sieves for $\mathrm{GL}_n$ and applications
本文为数域上 $\mathrm{GL}_n$ 的自守表示族建立了不依赖广义 Ramanujan 猜想的大筛法不等式，并应用于改进 $L$-函数均值界、零点密度估计及 Langlands 参数密度定理。
Improving on the Brun-Titchmarsh Theorem
研究通过结合大筛法和塞尔伯格筛法的混合筛法及‘局部模型’，证明当\(N\)足够大时，长度为\(N\)的区间内素数个数至多为\(2N/(\log N + 3.53)\)。
Large sieve inequality for sums of Legendre symbols over short intervals
本文利用Burgess界和Selberg筛法，研究Legendre符号在短区间上的二次矩界，并在特定条件下获得对h的幂次改进。
A sharp inverse theorem for the quadratic large sieve
研究二次逆大筛法问题，通过结合加权熵论证与筛法估计，改进相关下界，并证明依赖密度的变体，还结合其他定理表明素数分解中被加数与二次像有大交集。
We prove a bound on the number of primes with a given splitting behaviour in a given field extension. This bound generalises the Brun-Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to $x$.
By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from $\ell$-adic cohomology, as well as a Bombieri--Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke--Maass cusp form of trivial nebentypus, the eigenvalue of the $n$-th Hecke operator does not coincide with the Kloosterman sum $\mathrm{Kl}(1,n)$ for infinitely many squarefree $n$ with at most $100$ prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterm…
Kloosterman sign changes with moduli having at most five prime factors
研究具有至多五个素因子的模的克洛斯特曼符号变化，基于解析估计和优化筛法，用几何半权重改进方法，使相关系数降低，证明归一化克洛斯特曼和每个符号出现次数\(\gg X / \log X\)，改进了前人结果。
On the exponents of distribution of primes and smooth numbers
利用三重可分解权重，本文证明了素数与光滑数在算术级数中的均等分布，消除了对Selberg特征值猜想的依赖，并改进了孪生素数与连续光滑数计数的上界。
On the frequency of primes preserving dynamical irreducibility of polynomials
本文针对算术动力学中多项式迭代模素数不可约性的素数密度问题，通过使用 Selberg 筛法改进了以往基于 Brun 筛法的结果，在部分情况下实现了显著的定量提升。
Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatorname{SO}(n, 1)$
针对 $\operatorname{SO}(n,1)$ 中带尖点的几何有限薄子群，在临界指数 $\delta_\Gamma > 1/2$ 时建立了均匀谱隙，填补了 $n \geq 3$ 时 $\delta_\Gamma \in (1/2, n-2]$ 区域的空白。
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