Large Sieve Inequalities with Quadratic Amplitudes
Liangyi Zhao
本文建立了一个具有平方振幅的大筛法不等式。我们使用双重大筛子来建立非平凡界。
In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds.
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Large sieve inequalities for quartic characters
本文证明了四次Dirichlet特征标的一个大筛法不等式。结果类似于二次和三次Dirichlet特征的大筛不等式。
Remarks on the Bombieri-Davenport Large Sieve Inequalities
本文对特殊序列的 Bombieri-Davenport 大筛法不等式进行了改进。
Large Sieve Inequalities for Special Characters to Prime Square Moduli
本文对模为素数平方的一些特殊特征标建立了一个大筛法不等式。我们的结果给出了在一定范围内的非平凡估计。
Large Sieve Inequalities for Characters to Powerful Moduli
本文推广了Baier和Zhao的结果，得到了一个具有强模特征的大筛的一般公式，改进了Zhao的结果。
Amplification arguments for large sieve inequalities
利用放大思想，从相应的“调和”不等式出发，给出了“算术”大筛不等式的一个新证明。我们证明了这也适用于给出模形式的一些新的筛不等式，其中Hecke特征值被认为是整数模素数的约简的类似物。
Large Sieve Inequalities for Characters to Square Moduli
本文建立了一个具有平方模特征的大筛法不等式。人们期望这个结果比经典不等式弱，但至少在理论上弱不了多少。该方法可推广到更高的功率模。
Large Sieve Inequalities for periods of Maass forms
本文针对第一类Fuchsian群，建立了以偶权Maass形式双曲周期为权重的大筛法不等式，推广了Chamizo关于权0情形的工作，并旨在应用于双曲子群相关的计数问题。
We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.
In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.
Explicit quadratic large sieve inequality
本文得到了Heath-Brown二次特征大筛不等式的显式版本，并讨论了其在L-函数和二次域中的应用。
A quadratic large sieve inequality over number fields
给出并证明了数域上二次特征标的一个大筛法不等式。为此，我们引入了n阶Hecke族的概念。我们发展这些Hecke家庭的基本理论，包括版本的泊松求和公式。
An improved large sieve for quadratic characters via Hooley neutralisers and its applications
本文结合Hooley neutralisers与二次特征的大筛法，改进了该筛法，并成功应用于具有双曲高度条件的特征和问题。
We study the two-parameter quadratic sieve for a general test function. We prove, under some very general assumptions, that the function considered by Barban and Vehov [BV68] and Graham [Gra78] for this problem is optimal up to and including the second-order term. We determine that second-order term explicitly.
A large sieve inequality for characters to quadratic moduli
本文提出了一种针对二次模数的大型筛不等式，并应用该结果推导了与特定导数乘法字符相关的$ L $-函数扭系数的加权零密度估计。
Large sieve inequalities with power moduli and Waring's problem
本文通过将大筛法不等式与华林问题的受限变体关联，并利用经典除数界和 Wooley 的指数和均值结果，改进了所有 k≥4 情形下带 k 次幂模的大筛法不等式，同时给出了基于 Hardy-Littlewood 猜想公式的条件性结果。
Spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i])
对SL（2，Z[i]）的任意Hecke同余子群Γ=Γ_0（q_0），我们证明了SL（2，C）上的Γ-自守尖点形式的含傅里叶系数和的一些新的上界（或“谱大筛不等式”）.所讨论的傅里叶系数可以由在任意给定的r的尖点c处的傅里叶展开式产生：我们的结果不限于c是无穷远处的尖点的情况。由于这个原因，我们的证明依赖于Hristina Lokvenec-Guleska在她的博士论文中得到的关于Γ \SL（2，C）的谱-Kloosterman求和公式的任意尖点的推广（在几个方面推广了Roelof Bruggeman和Yoichi Motohashi关于PSL（2，Z[i]）\PSL（2，C）的求和公式，尽管不是关于尖点的选择）。证明所需的扩展的总和公式是在附录中。
Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$
本文通过证明例外Maass形式的新型大型筛不等式，改进了n²+1的最大大素因子的下限，并应用于素数和光滑数在算术级数中的分布指数。
We develop a version of Ekedahl's geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.
The Large Sieve Inequality for Quadratic Polynomial Amplitudes
本文对赵良义提出的一个二次多项式振幅的大筛不等式作了适当的改进。
Optimization of Quadratic Sieve Algorithm Implementation for Large Integer Factorization
本文提出二次筛算法的两种优化：根值累加改为mdi以降低复杂度至O(n/m)，参数查找改用二分查找降至O(logn)，从而提升RSA模数分解效率。
Smooth Subsum Search: A heuristic for practical integer factorization
本文提出一种基于可被因子基素数整除的和来表示平滑性候选的整数分解启发式方法，其Python实现比最快的自初始化二次筛法快5至10倍。
Inverse questions for the large sieve
设对每个充分大的素数$p$，无限集$A$至多占有模$p$的剩余类$\frac{1}{2}(p+1)$.平方，或者更一般地，任何二次的整数值，是这样的集合的一个例子。通过大筛法不等式，证明了$A$中最多为$X$的元素个数为$O(X^{1/2})$，二次型例子表明了这一点.逆大筛问题的最简单形式是询问它们是否是唯一的例子。我们证明了各种结果，并制定了与此问题有关的各种解决方案，包括几个改进的大筛界时，所占用的剩余类$A$有一些加法结构。不幸的是，我们无法解决问题本身。
Weighted spectral large sieve inequalities for Hecke congruence subgroups of SL(2,Z[i])
本文证明了关于群SL（2，Z[i]）的Hecke同余子群Γ=Γ_0（q）自守的尖点形式傅里叶系数和的加权平均值的新界，并对应于空间L^2（Γ \SL（2，C）/SU（2））上Laplace算子的例外特征值.这些结果是，对于某些应用，一个有效的替代广义Selberg特征值猜想。我们给出一个这样的应用程序的证明，这是一个上界的总和广义Kloosterman和（的意义，在研究某些平均值的Hecke zeta函数与groessencharakters）。我们的证明广泛使用了Lokvenec-Guleska对PSL（2，Z[i]）\PSL（2，C）的Bruggman-Motohashi求和公式的推广.我们还采用了金和Shahidi的有关拉普拉斯算子的第一特征值的界，和一个“未加权”谱大筛不等式（我们的证明，这是单独出现）。
On the Twisted Spectral Large Sieve Inequality for $\mathrm{PGL}_2 (\mathbb{Z}[i]) \backslash \mathrm{PGL}_2 (\mathbb{C})$
本文将 Deshouillers 等人的扭曲谱大筛不等式从 $\mathrm{SL}_2 (\mathbb{Z})$ 扩展至 $\mathrm{PGL}_2 (\mathbb{Z}[i])$。
We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri--Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes $p$such that $p-1$ has a prime divisor of size $\gg p^{2/5+o(1)}$ that is the value of an incomplete norm form polynomial.
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。
The large sieve for self-dual Eisenstein series of varying levels
本文证明了变水平自对偶 Eisenstein 级数的最优大筛不等式，可解释为有理数按高度排序的大筛，方法递归且与已知大筛法有共同点。
We study the sum $Σ_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{μ(d)μ(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, so that a continuous, monotonic and explicit version of Selberg's sieve can be stated. Thanks to Barban-Vehov (1968), Motohashi (1974) and Graham (1978), it has been long known, but never explicitly, that $Σ_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $Σ_q(U)\sim\frac{q}{φ(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $Σ_q(U)$, a constant $\mat…
Surviving Eratosthenes sieve I: quadratic density and Legendre's conjecture
本文研究了埃拉托色尼筛法中二次密度的概念，探讨了素数间隙在不同阶段的生存情况及Legendre猜想的预测能力。
We improve Irving's method of the double-sieve by using the DHR sieve. By extending the upper and lower sieve functions into their respective non-elementary ranges, we are able to make improvements on the previous records on the number of prime factors of irreducible polynomials at prime arguments. In particular, we prove that irreducible quadratics over $\mathbb{Z}$ satisfying necessary local conditions are $P_4$ infinitely often.
CUDA-MPQS: A GPU-Resident Self-Initializing Quadratic Sieve, and the Factorization of RSA-155
提出CUDA-MPQS，一种全阶段在GPU上执行的自初始化二次筛法，成功分解RSA-155，创下二次筛法最大整数纪录，并在RSA-100上显著快于CPU实现。
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。
The large sieve inequality is equivalent to the bound $λ_1 \leqslant N + Q^2-1$ for the largest eigenvalue $λ_1$ of the $N$ by $N$ matrix $A^{\star} A$, naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is $N \asymp Q^2$. Based on his numerical data Ramaré conjectured that when $N \sim αQ^2$ as $Q \rightarrow \infty$ for some finite positive constant $α$, the limiting distribution of the eigenvalues of $A^{\star} A$, scaled by $1/N$, exists and is non-degenerate. In this paper we prove this conjecture …
Motivated by a desire to understand the distribution of roots of cubic congruences, we re-derive a parametrization of roots $ν\pmod m$ of $X^3 \equiv 2 \pmod m$ found by Hooley. Although this parametrization does not lead us here to anything towards proving equidistribution of the sequence $\fracν{m} \pmod 1$, we are able to prove spacing results, and then a large sieve type inequality, which we view as analogous to the large sieve inequality for roots of quadratic congruences found by Fouvry and Iwaniec in their proof that there are infinitely many primes of the form $n^2 + p^2$. The parametr…
We significantly improve our results of Glas. Mat., III. Ser. 53(2018), No. 2, 229-238, reducing relative Thue inequalities to absolute ones.
A spectral resolution of the large sieve
本文利用单参数核差分算子族的特征值，对 $Q\ge N^{7/8}$ 时的大筛法二次型进行完整谱分析，并证明在 $Q\asymp N$ 时其可在小维数空间上偏离期望的 Riemann 和行为。
Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities \[ |F(x,y)|\leq K \;\; (x,y\in Z_M) \] to the resolution of absolute Thue inequalities of type \[ |F(x,y)|\leq k \;\; (x,y\in Z). \] We illustrate our method with an explicit example.
Computing points on bielliptic modular curves over fixed quadratic fields
本文提出 Mordell-Weil 筛，用于计算固定二次域上若干双椭圆模曲线 $X_0(N)$ 的点，并考察八个水平及 $|d|<100$ 的二次域。
We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form $q(x_1,\cdots,x_n)=m$, where $q$ is a non-degenerate integral quadratic form in $n\geqslant 3$ variables and $m$ is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.
For any given non-square integer $ D\equiv 0,1 \pmod{4} $, we prove Euclid's type inequalities for the sequence $ \{q_{i}\} $ of all primes satisfying the Kronecker symbol $ (D/q_{i})=-1 $, $ i=1,2,\cdots, $ and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones's argument in the classification of regular ternary quadratic forms to some extent.
We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.
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。
Decoupling for two quadratic forms in three variables: a complete characterization
该研究证明了ℝ⁵中三变量两个二次型对应的余维数2退化曲面的精确解耦不等式，结合已有非退化情形的研究，完成了三变量二次型对的解耦不等式分类。
We prove the existence of S-integral solutions of simultaneous diophantine inequalities for pairs (Q,L) involving one quadratic form and one linear form satisfying some arithmetico-geometric conditions. The proof uses strong approximation in algebraic groups and Ratner's topological rigidity of unipotent actions on homogeneous spaces.
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