A Large Sieve Inequality for Euler Products

显式二次大筛不等式

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 new bound for the large sieve inequality with power moduli

We give a new bound for the large sieve inequality with power moduli q^k that is uniform in k. The proof uses a new theorem due to T. Wooley from his work on efficient congruencing.

Euler乘积筛法

Euler Product Sieve

本文研究了基于Euler乘积的Riemann zeta函数逼近，证明了Euler Product Sieve生成素数的合理性，并指出Bounded Riemann Hypothesis等价于Riemann Hypothesis与零点简单性的合取。

The large sieve inequality with square moduli for quadratic extensions of function fields

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.

A large sieve inequality of Elliott--Montgomery--Vaughan type for Maass forms on GL(n,R) with applications

In this paper, we establish a large sieve inequality of Elliott-Montgomery-Vaughan type for Maass forms on GL(n, R) and explore three applications.

A quadratic large sieve inequality over number fields

We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

二次模数的大型筛不等式

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。

Amplification arguments for large sieve inequalities

We present a new proof of the "arithmetic" large sieve inequality, starting from the corresponding "harmonic" inequality, which is based on an amplification idea. We show that this also adapts to give some new sieve inequality for modular forms, where Hecke eigenvalues are thought as the analogues of the reductions of integers modulo primes.

Large sieve inequality with power moduli for function fields

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high dimensions and for power moduli.

关于 $\mathrm{PGL}_2 (\mathbb{Z}[i]) \backslash \mathrm{PGL}_2 (\mathbb{C})$ 的扭曲谱大筛不等式

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 $\mathrm{PGL}_2 (\mathbb{Z}[i])$。

大筛法不等式与平方和的加法分解

The large sieve inequality and additive decompositions of sums of squares

该研究基于大筛法证明了针对缺失素数模平方剩余类集合的新筛法估计，将其应用于Ostmann问题，得出平方和的非平凡和集分解需由大小相近的两个集合构成的结论。

通过Parseval的加权大筛法

The weighted large sieve through Parseval

本文通过Parseval恒等式而非近似Bessel不等式改进大筛法的算术形式，从而在加权大筛法不等式上取得超越先前方法启发式界限的改进，并讨论了该方法的优化性。

Large Sieve Inequalities for Characters to Square Moduli

In this paper, we develop a large sieve type inequality with characters to square moduli. One expects that the result should be weaker than the classical inequality, but, conjecturally at least, not by much. The method is generalizable to higher power moduli.

The Large Sieve Inequality for Quadratic Polynomial Amplitudes

We provide here a modest improvement upon a large sieve inequality for quadratic polynomial amplitudes orginally due to Liangyi Zhao.

On the exponential large sieve inequality for sparse sequences modulo primes

We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $λ^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.

关于 $\mathrm{PGL}_2 (\mathbb{Z}[i]) \backslash \mathrm{PGL}_2 (\mathbb{C})$ 的谱大筛不等式

On the Spectral Large Sieve Inequality for $\mathrm{PGL}_2 (\mathbb{Z}[i]) \backslash \mathrm{PGL}_2 (\mathbb{C})$

本文引入复Bessel积分的简明表达式，改进了Watt关于PGL_2(Z[i])\PGL_2(C)的谱大筛不等式，并在谱参数平均值位于以(0,0)为中心的正方形时取得理想结果。

非阿贝尔大筛法与阿廷原根猜想

A non-abelian large sieve and Artin's primitive root conjecture

本文研究经典阿贝尔大筛法不等式的非阿贝尔推广，假设该推广成立可证明阿廷原根猜想，还通过对偶技术得到针对该推广的无条件结果。

The large sieve with square norm moduli in Z[i]

We prove a large sieve inequality for square norm moduli in Z[i].

一个与埃拉托斯特尼筛法相关的不等式

An inequality related to the sieve of Eratosthenes

本文研究埃拉托斯特尼筛法相关的计数函数 $Φ(x,y)$，证明了除少数小情形外，当 $y\le\sqrt{x}$ 时该函数满足严格上界 $Φ(x,y)<.6x/\log y$。

通过Parseval的加权大筛法

The weighted large sieve through Parseval

本文通过使用Parseval恒等式改进加权大筛不等式，并计算了Beurling-Selberg优超函数的L2范数。

Large sieve inequalities for quartic characters

In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.

变水平自对偶 Eisenstein 级数的大筛法

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。

Asymptotic Large Sieve

Motivated by applications to the study of L-functions, we develop an asymptotic version of the large sieve inequality for linear forms in primitive Dirichlet characters.

Large Sieve Inequalities with Quadratic Amplitudes

In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds.

Spacing and A Large Sieve Type Inequality for Roots of a Cubic Congruence

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…

Luo的谱大筛法不等式在短区间上的推广

Luo's Spectral Large Sieve Inequality on Short Intervals

本文在短区间上推广Luo的谱大筛法不等式，证明Eisenstein-Kloosterman相消在$T<t_j\le T+M$（$\sqrt{T}<M\le T$）有效，并改进原不等式。

Large Sieve Inequalities for Special Characters to Prime Square Moduli

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

Limiting distribution of eigenvalues in the large sieve matrix

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 …

The large sieve with power moduli for $\mathbb{Z}[i]$

We establish a large sieve inequality for power moduli in $\mathbb{Z}[i]$, extending earlier work by L. Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in $\mathbb{Z}$. Our method starts with a version of the large sieve for $\mathbb{R}^2$. We convert the resulting counting problem back into one for $\mathbb{Z}[i]$ which we then attack using Weyl differencing and Poisson summation.

Primes in short arithmetic progressions

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.

Additive energy and a large sieve inequality for sparse sequences

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K.~Halupczok (2012, 2015, 2018) and M.~Munsch (2020). We also consider moduli defined by polynomials $f(X) \in \mathbb{Z}[X]$, Piatetski-Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri--…

Legendre符号在短区间上的大筛不等式

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。

On an inverse ternary Goldbach problem

We prove an inverse ternary Goldbach-type result. Let $N$ be sufficiently large and $c>0$ be sufficiently small. If $A_1,A_2,A_3\subset [N]$ are subsets with $|A_1|,|A_2|,|A_3|\geq N^{1/3-c}$, then $A_1+A_2+A_3$ contains a composite number. This improves on the bound $N^{1/3}$ from Gallagher's larger sieve. The main ingredients in our argument include a type of inverse sieve result in the larger sieve regime, and a variant of the analytic large sieve inequality.

The principle of the large sieve

We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We get some first results on the number of prime divisors of ``most'' elements of an elliptic divisibility sequence, and we develop in some detail ``probabilistic'' sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some st…

On the distribution of fractions with power denominator

In this paper we obtain a sharp upper bound for the number of solutions to a certain diophantine inequality involving fractions with power denominator. This problem is motivated by a conjecture of Zhao concerning the spacing of such fractions in short intervals and the large sieve for power modulus. As applications of our estimate we show Zhao's conjecture is true except for a set of small measure and give a new $\ell_1 \rightarrow \ell_2$ large sieve inequality for power modulus.

几何格的布伦不等式

Brun's inequality for a geometric lattice

本文在几何格上证明了布伦不等式并发展相应筛法，利用拟阵惠特尼数对数凹性，并研究道林格惠特尼数移位卷积及广义道林数渐近公式。

Character sums with smooth numbers

We use the large sieve inequality for smooth numbers due to S. Drappeau, A. Granville and X. Shao (2017), together with some other arguments, to improve their bounds on the frequency of pairs $(q,χ)$ of moduli $q$ and primitive characters $χ$ modulo $q$, for which the corresponding character sums with smooth numbers are large.

一个与欧拉-黎曼ζ函数、digamma函数和单位区间上余切函数相关的简单不等式

A simple inequality relating the Euler-Riemann zeta function, digamma, and cotangent over the unit interval

本文证明了一个与欧拉-黎曼ζ函数、digamma函数和余切函数相关的简单不等式，并提出猜想性加强及问题起源的评论。

Motivic Euler products in motivic statistics

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's sieve, including work of Bucur-Kedlaya on complete intersections and Erman-Wood on semi-ample Bertini theorems. A key ingredient is the use of motivic Euler products, as introduced by the first au…

Almost all elliptic curves are Serre curves

Using a multidimensional large sieve inequality, we obtain a bound for the mean square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove a theorem to the effect that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.

论线性递推序列的 Euler 函数

On the Euler function of linearly recurrence sequences

本文研究整数线性递推序列通项与其 Euler 函数之间的不等式，证明对非退化且非多项式通项的序列，该不等式在密度 1 的整数上成立，并给出失效集合的计数上界。

Prime and zero distributions for meromorphic Euler products

The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution.

On Nicolas criterion for the Riemann Hypothesis

Nicolas criterion for the Riemann Hypothesis is based on an inequality that Euler totient function must satisfy at primorial numbers. A natural approach to derive this inequality would be to prove that a specific sequence related to that bound is strictly decreasing. We show that, unfortunately, this latter fact would contradict Cramér conjecture on gaps between consecutive primes. An analogous situation holds when replacing Euler totient by Dedekind $Ψ$ function.

Euler's Function on Products of Primes in Progressions

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $φ(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(e^{2πi/q})$ is true if and only if for all integers $k\geq 1$ we have \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,1)}.\] Here $\bar{N}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equ…

On primitive Dirichlet characters and the Riemann hypothesis

For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-γ} ϕ(2n_k)/\log\log(2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $γ$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $ϕ(n)$ is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many $k$ for which the same inequality holds and infinitely many $k$ for which it fails to hold.

关于一个Nicolas不等式的验证

On the verification of a Nicolas inequality

本文研究Nicolas不等式，指出其验证依赖于Mertens估计中误差项的符号，并分析了该误差项的符号。