Harald Helfgott
We improve on the best available bounds for the square-free sieve and provide a general framework for its applicability. The failure of the local-to-global principle allows us to obtain results better than those reached by a classical sieve-based approach. Techniques involving sphere-packing yield upper bounds on the number of integer and rational points on curves of positive genus.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give a short alternative proof using Heath-Brown's square sieve of a bound of the author for the large sieve with square moduli.
We prove a large sieve inequality for square norm moduli in Z[i].
The large sieve for square moduli, revisited
本文在假设模平方根的更高加法能量条件下，改进了平方模的大筛法。
We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.
We prove an estimate for the large sieve with square moduli which improves a recent result of L. Zhao. Our method uses an idea of D. Wolke and some results from Fourier analysis.
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.
We prove a lower bound for the large sieve with 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.
We establish a result on the large sieve with square moduli. These bounds impro ve recent results by S. Baier(math.NT/0512228) and L. Zhao(math.NT/0508125).
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.
We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound obtained in arXiv:1802.03131 by Baier and Singh. Indeed, we point out an error in arXiv:1802.03131.
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。
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。
We establish a general large sieve inequality with sparse sets $\mathcal{S}$ of moduli in the Gaussian integers which are in a sense well-distributed in arithmetic progressions. This extends earlier work of S. Baier on the large sieve with sparse sets of moduli. We then use this result to obtain large sieve bounds for the cases when $\mathcal{S}$ consists of squares of Gaussian integers and of Gaussian primes. Our bound for the case of square moduli improves a recent result by the authors.
We formulate a general problem: given projective schemes $\mathbb{Y}$ and $\mathbb{X}$ over a global field $K$ and a $K$-morphism $η$ from $\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in $\mathbb{X}(K)$ of height at most $B$ have a pre-image under $η$ in $\mathbb{Y}(K)$? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a non-trivial answer to the general problem when $K=\mathbb{F}_q(T)$ and $\mathbb{Y}$ is a prime degree…
Partial progress towards the large sieve for square moduli
本文基于Bag和Shparlinski的方法，研究了双线性和在s=1/2时的情况，探讨了模平方根的存在性，并回顾了相关双线性和的研究成果。
Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then use this result together with Fourier techniques to obtain large sieve bounds for the case when S consists of squares. These bounds improve a recent result by L. Zhao.
Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$. We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} = \mathbb{Q}( π_p )$ of the reduction of $A$ modulo primes $p$ at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which this "Frobenius field" is a specified number field and, when $A$ is two-dimensional, at which the Frobenius field contains a specified real quadratic number fi…
Almost primes and primes that are sums of two squares plus 1
该研究用向量筛框架（含半线性筛、线性筛）得出满足p-1为两平方和且p+2素因子个数有界的素数p≤x的下界。
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $p \neq \ell$, the Frobenius automorphism associated to $p$ (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of $\ell$. Its splitting field is called the Frobenius field of $E$ at $p$. Let $E_1$ and $E_2$ be two elliptic curves defined over $\mathbb{Q}$ that are non-isogenous over $\overline{\mathbb{Q}}$ and also without complex multiplication over $\…
On the Ekedahl sieve for the singular locus of the discriminant polynomial
针对判别式多项式奇异点，提出一种改进的Ekedahl筛法变体，通过利用判别式非退化性质简化筛法步骤，获得更优的误差项和模量范围。
A matching lower bound for the sextic large sieve and structured prime moments
针对艾森斯坦整数上的六次大筛法，在指标均限制为无平方理想的不平衡范围内，建立了匹配的精确下界，解决了无平方模下保留高斯和提供的二阶矩的难点，证明了无平方投影保留原矩的正比例并将下界转移至无平方行。
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.
Symmetric Square Large Sieve for $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $ and Prime Geodesic Theorem for $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $
本文针对PSL₂(Z[i])\H³的素测地线定理误差项，通过建立PSL₂(Z[i])\PSL₂(C)上对称平方的谱大筛不等式，将对称平方L-函数二阶矩的界从O(T^{3+2/3+ε})改进为O(T^{3+1/2+ε})。
Optimal large sieve for fixed order characters
本文针对固定阶 Hecke 特征改进大筛法不等式，得到所有参数范围内最优的界，并应用于三次特征第二矩的紧上界、渐近式及矩和零点密度估计。
In this paper we study the distribution of squares modulo a square-free number $q$. We also look at inverse questions for the large sieve in the distribution aspect and we make improvements on existing results on the distribution of $s$-tuples of reduced residues.
We introduce a variant of the large sieve and give an example of its use in a sieving problem. Take the interval [N] = {1,...,N} and, for each odd prime p <= N^{1/2}, remove or ``sieve out'' by all n whose reduction mod p lies in some interval I_p of Z/pZ of length (p-1)/2. Let A be the set that remains: then |A| << N^{1/3 + o(1)}, a bound which improves slightly on the bound of |A| << N^{1/2} which results from applying the large sieve in its usual form. This is a very, very weak result in the direction of a question of Helfgott and Venkatesh, who suggested that nothing like equality can occu…
Sarnak's Program for Erdős Sieves. Part I: Topological Dynamics and Light Tails
本文扩展了Sarnak的计划到Erdős筛子，通过定义轻尾条件并证明其与Mirsky测度下的R-自由数集的普遍性相关。
$ $The aim of this thesis is to lower the bound on square-free primitive roots modulo primes. Let $g^{\Box}(p)$ be the least square-free primitive root modulo $p$. We have proven the following two theorems. Theorem 0.1. $$g^{\Box}(p) < p^{0.88}\quad\text{for all primes }p.$$ Theorem 0.2. $$g^{\Box}(p) < p^{0.63093}\quad\text{for all primes } p < 2.5\times10^{15} \text{ and }p > 9.63\times10^{65}.$$ Theorem 0.1 shows an improvement in the best known bound while Theorem 0.2 shows for which primes we can prove the theoretical lower bound. After some introductory information in Chapter 1, we will …
On certain bilinear sums with modular square roots and applications
扩展了Dunn等人关于模平方根加法能量的界，并将结果应用于带模平方根的双线性指数和，在平方模大筛法上取得部分进展。
Distribution of square-free palindromes
研究平方自由回文数的分布，通过回文数在算术进展中的上界估计，结合指数和估计、6近似质回文数研究及大筛法，建立分布指数为1/16的结论。
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 $Φ(x,y)<.6x/\log y$。
Almost primes between all squares
本文证明了对于所有 $n\geq 1$，在 $n^2$ 和 $(n+1)^2$ 之间存在至多有 4 个素因子的数，这是首个对所有 $n$ 成立的此类结果，方法依赖 Sorenson 和 Webster 的计算、显式线性筛法及新证明的显式 Kuhn 加权筛法。
We prove that the average size of the squares of differences between consecutive primes less than $x$ is $O(x^{0.23+\varepsilon})$ for any fixed $\varepsilon>0$. This improves on a result of Peck, who gave bound $O(x^{0.25+\varepsilon})$ in the place of $O(x^{0.23+\varepsilon})$. Key ingredients are Harman's sieve, Heath-Brown's mean value theorem for sparse Dirichlet polynomials and Heath-Brown's $R^*$ bound.
The least prime with a given primitive root and a variant of the larger sieve
本文在广义黎曼猜想下，将给定原根的最小素数上界改进为对数三次方量级，并证明大筛法新变体，优于现有结果。
For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert αn + β\rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In addition, we also establish an asymptotic formula with an additional square-free condition on $n$. Moreover, if $α$ is quadratic irrational then the asymptotic formulas holds for all sufficiently large $x$. Our ingredients come from the Harman sieve which we adapt suitably to sieve for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for exponential s…
Sieving with square conditions and applications to Hilbert cubes in arithmetic sets
本文探讨了Gallagher筛法在模素数平方下的应用局限，并提出通过转换余数信息以提升筛法效果，同时研究了Hilbert立方体在不同乘法定义集中的最大维度，得出了相关上界。
Weighted enumeration of number-fields and counting points that take bounded squarefree values along certain polynomials using Pseudo and Sudo maximal orders
利用Ekedhal筛法计数与判别式多项式密切相关的多项式的有界无平方因子值，并得到判别式有界数域数量的加权下界，其limsup远大于当前已知界。
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。
An improvement of the large sieve for square moduli under Hooley's hypothesis $R^*$ for short Salié sums
在Hooley假设$R^*$下，将平方模数大筛法在临界点$N=Q^3$的界改进为$Q^{1/2-1/134+\varepsilon}$，通过精确计算高斯和将模平方根和转化为Salié和，获得对所有模数有效的平方根抵消。
Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\dots,x\}$. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event, and conjectured that it in fact exhibits a sharp threshold in $N$. More recently,…
Representations of positive integers by three almost-prime squares
本文利用高维筛法和Richert型加权筛法，证明了满足特定同余条件的大整数可表示为三个平方和，且乘积或每个平方根为几乎素数，改进了Waibel和Banerjee的结果。
We provide a new proof of Hua's result that every sufficiently large integer N\equiv 5(mod\,24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of $L$-functions. Here, we present a proof based on the transference principle first introduced by Green. Using a sieve theoretic approach similar to the work by Shao, we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudorandomness of the majorant of the characteri…
We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions $a, b \in \mathbb N$ to the equations $a+b=n$ and $a-b=n$, where $a$ is $k$-free and $b$ is $l$-free. This is the first time that this problem has been studied with distinct powers $k$ and $l$.
We apply a variant of the square-sieve to produce a uniform upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over the projective line, whose general fibre is a hyperelliptic curve.
Primes in arithmetic progressions and short intervals without $L$-functions
本研究提出一种可检测乘法结构集合中素数的筛法，给出了Linnik最小素数问题和短区间素数存在性的无$L$-函数新证明，为相关数论问题提供了新的研究工具。
On the Moments of the Number of Representations as Sums of Two Prime Squares
本文研究整数表为两个素数平方之和的表示数的各阶矩，利用 Selberg 筛法获得无条件上界，并在 Green-Tao 定理推广的条件下证明下界，且从第五阶矩起得到预期数量级。
We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{θ/2}$, for some fixed $θ< 1$. When $s = 5$ we use a sieve method to show that all sufficiently large integers $n \equiv 5 \pmod {24}$ can be represented in the above form for $θ> 8/9$. This improves on earlier work by Liu, Lü and Zhan, who established a similar result for $θ> 9/10$. We also obtain estimates for the number of integers $n$ satisfying the necessary local conditions but lacking representations of the above form with $s = 3, 4$. When $…
First occurrences of square-free gaps and an algorithm for their computation
本文使用基于埃拉托色尼筛法的算法搜索无平方数间隙的首次出现，并报告了长度为10至18的间隙的起始数值。
A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^α$-smoothness. Using bounds on multiplicative character sums an…
We generalise the square sieve developed by Heath-Brown to higher powers in order to improve on the error term for the problem of counting consecutive power-free numbers.
A prime plus a square: A conditional 11/20 upper bound
本文在本原Dirichlet L函数非平凡零点实部≤7/8的假设下，结合导体加权均方估计等方法，得到不能表示为素数加平方数的正整数计数的11/20上界。
Sieve dimension and search depth for the Erdős-Straus conjecture, $n \equiv 1 \pmod{24}$
本文针对模24余1的素数，研究Erdős-Straus猜想的无因式分解搜索深度，证明了相关集合的上界，展示了无条件障碍，还提出了可判定区间内素数猜想的无因式分解方法，该方法存在I/II型遍历的代价不对称性。
A combinatorial large sieve for Sidon sets, distances, and norm forms
提出一种针对有界代数重数集合的组合大筛法，通过模小素数分裂产生碰撞与全局有界性矛盾，证明Sidon平方集、无重复距离网格点集等问题的超多对数上界。
A Lock-Free, Fully GPU-Resident Architecture for the Verification of Goldbach's Conjecture
本文提出一种无锁、完全驻留于GPU的架构，通过优化内存管理和异步工作窃取池，实现高效验证哥德巴赫猜想，达到高并行效率和算法加速。
We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1/ζ(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves …
An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes
本文给出 Bombieri 无对数密度估计的显式版本，并结合 Green 的工作证明：若集合 A⊆{1,…,N} 中任意两元素之差不为 p−1（p 为素数），则 |A|≪N^{1−1/10^{18}}，从而加强了 Sárközy 定理。
Let $g^{\square}(p)$ denote the least square-free primitive root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$.
Counting square-free values of random polynomials
本文研究了随机多项式平方自由值的计数问题，证明了平均误差项与主项的四次方根关系。
On the hyperbolic prime number theorem
本文针对双曲素数定理，研究无平方因子距离的渐近计数，无条件证明主项阶为 x，并利用加权线性筛得到至多 7 个素因子时的下界，同时推广到更一般的分布假设。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。