Stephan Baier
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.
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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。
In this note we give a new bound for large sieve with characters to power moduli which improves in some range of the parameters the previous bounds of Baier/Zhao and Halupczok.
We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.
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 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.
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 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 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.
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 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 aim to generalize the results in Baier and Zhao and develop a general formula for large sieve with characters to powerful moduli that will be an improvement to the result of Zhao.
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.
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.
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 apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao.
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。
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.
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。
Primes in arithmetic progressions to large moduli and refinements of Harman's sieve
利用多作者算术信息与Harman筛法变体，构造素数指示函数的上下界，得到模数达x^{9/17}的双线性形式和模数达x^{17/32}的三线性形式的均值定理，并给出π(x;q,a)的几乎全部模数q的新上下界。
We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was log p to an exponent tending to infinity with p, lies in the use of the linear sieve (a particular version called the shifted sieve) rather than Brun's sieve. The same methods allow us to rederive a conditional result of Shoup on the least prime primitive root (mod p) for all prime moduli p, assuming the generalized Riemann hypothesis. We also extend both res…
We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups.
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。
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。
We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.
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.
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--…
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。
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。
A modification of the linear sieve, and the count of twin primes
本文提出一种满足强分解性质的线性筛法修改，将素数分布水平提升至 $x^{10/17}$，超越以往工作，并据此获得孪生素数计数新上界，实现自1986年以来最大改进。
Exceptional characters and prime numbers in sparse sets
本文在假设有无穷多个例外特征的前提下提出素数下界筛法，以更少算术信息获得非平凡下界，并据此证明形如 $a^2+b^8$ 的素数有无穷多个。
Horizontal moments of Kloosterman sums and a density version of sieves
本研究针对模$n$的标准Kloosterman和，研究其关于$n\leqslant X$的平均矩，得到了和的水平矩渐近估计，通过构造特殊模族结合筛法密度版本与Sato--Tate分布完成上下界证明。
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。
In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form $p=am^2+1$ such that $a\leq p^{5/9+ε}$.
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。
Bilinear forms with Kloosterman sums via quadratic characters
研究带有克洛斯特曼和的双线性形式，基于与二次特征和的新联系证明新界限，在临界范围有改进，还给出在扭曲L函数矩及特殊马阿斯形式大筛法中的应用。
Non-abelian amplification and bilinear forms with Kloosterman sums
利用SL_2(Z/cZ)上的傅里叶分析和非阿贝尔特征放大，对复合模Kloosterman和的双线性和给出非平凡界，改进所有模的Pólya-Vinogradov范围，并应用于扭尖L-函数矩和大筛法不等式。
Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the number of such points of height $\leq B$ grows slower than any positive power of $B$. For example, homogeneous integral polynomials in a fixed number of variables and degree, with discriminant divisible only by a fixed set of primes, are sparse when considered up to integral linear substitutions.
A Bombieri-Vinogradov-type theorem for moduli with small radical
本文将关于素数幂稀疏集的 Bombieri-Vinogradov 型定理推广至具有小根的模的稀疏集，通过结合先前方法与 Roger Baker 的一般模结论得出结果。
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