Liangyi Zhao
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.
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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.
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.
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.
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.
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。
In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds.
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.
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.
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 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.
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.
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。
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…
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 prove, in respect of an arbitrary Hecke congruence subgroup Γ=Γ_0(q_0) of the group SL(2,Z[i]), some new upper bounds (or `spectral large sieve inequalities') for sums involving Fourier coefficients of Γ-automorphic cusp forms on SL(2,C). The Fourier coefficients in question may arise from the Fourier expansion at any given cusp c of Γ: our results are not limited to the case in which c is the cusp at infinity. For this reason, our proof is reliant upon an extension, to arbitrary cusps, of the spectral-Kloosterman sum formula for Γ\SL(2,C) obtained by Hristina Lokvenec-Guleska in her doctor…
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 prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a Hecke congruence subgroup Γ=Γ_0(q) of the group SL(2,Z[i]), and correspond to exceptional eigenvalues of the Laplace operator on the space L^2(Γ\SL(2,C)/SU(2)). These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a proof of one such application, which is an upper bound for a sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-function…
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--…
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 $\mathrm{PGL}_2 (\mathbb{Z}[i])$。
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。
We prove that the Kloosterman sum $S(1,1;c)$ can change sign infinitely often as $c$ runs over squarefree moduli with at most 10 prime factors, which improves the previous results of E. Fouvry and Ph. Michel, J. Sivak-Fischler and K. Matomäki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms.
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。
We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem to this setting, we apply the polynomial large sieve inequality that has been recently proved and which includes recent progress in Vinogradov's mean value theorem due to Parsell et al. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.
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。
In this paper we give a full description of the inequalities that can occur between overpartition ranks. If $ \overline{N}(a,c,n) $ denotes the number of overpartitions of $ n $ with rank congruent to $ a $ modulo $ c,$ we prove that for any $ c\ge7 $ and $ 0\le a<b\le\left\lfloor\frac{c}{2}\right\rfloor $ we have $ \overline{N}(a,c,n)>\overline{N}(b,c,n) $ for $n$ large enough. That the sign of the rank differences $ \overline{N}(a,c,n)-\overline{N}(b,c,n) $ depends on the residue class of $ n $ modulo $ c $ in the case of small moduli, such as $ c=6, $ is known due to the work of Ji, Zhang a…
The large sieve for square moduli under Hooley's hypothesis $R^*$
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。
Explicit inequalities for the nth lucky number
本文基于幸运数分布特性，推导出第n个幸运数的显式上下界，类比素数定理揭示其渐进行为。
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.
On bilinear sums with modular square roots and applications III
本文改进了之前方法，针对素数平方模，通过限制二次高斯和到简化剩余类，实现显著消去，提升双线性和的分析。
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。
Explicit Burgess inequalities for cubefree moduli
本文改进了r=2时的显式Burgess不等式常数，并为立方自由模数q的r≥3情况提供了显式界。
In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simultaneous inequalities we require at least $m+2$ variables, improving upon existing methods, which generically require at least $2m+1$ variables. Our result also generalises the theorem of Green-Tao-Ziegler on linear equations in primes. Many of the methods presented apply for arbitrary coefficients, not just for algebraic coefficients, and we formulate a conjecture concerning the pseudorandomness of sieve weights which…
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。
Sign changes of Kloosterman sums with moduli having at most six prime factors
本文通过引入新的截断除数函数，改进了克洛斯特曼和符号变化的研究结果，结合了多种数学工具。
We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square free moduli and also on the result due to Gallagher and Iwaniec when the core $q'=\prod_{p|q}p$ of the modulus $q$ satisfies $\log q'\sim \log q$. Some applications to zero free regions of Dirichlet L-functions and the $\rm{P\acute{o}lya}$ and Vinogradov inequalities are indicated.
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^θ$ for $θ=1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $θ=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $θ=4/9-\varepsilon.$ Similarly as in the work of Matomäki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $θ=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as …
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。
In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose cardinality is bounded by a power of $\log x$. We use a $p$-adic variant Harman's sieve to extend Baker's range to $q\le x^{1/4-\varepsilon}$ if $\mathcal{S}$ is restricted to prime powers $p^N$, where $p\le (\log x)^C$ for some fixed but arb…
Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier
本文利用三重可良好分解权重将素数分布水平提升至0.617，并在假设Selberg猜想下达到0.625，由此为孪生素数和Goldbach问题取得新上界，实现了Goldbach问题自1966年以来的最大改进。
On the largest prime factors of $p-1$ and $p+1$
本文证明对正比例的素数，$p-1$ 与 $p+1$ 的最大素因子大小关系在两个方向均成立，方法结合对数平衡恒等式与均匀塞尔伯格筛法。
Rudin Inequality, Chang Theorem, primes and squares
本文证明素数密度子集上三角多项式大值具有较弱加性结构，并通过对区间、素数及平方数上的解离集建立大筛法不等式，推广了 Rudin 不等式与 Chang 定理。
An explicit lower bound for large gaps between some consecutive primes
本文确定了相邻素数间大间隙下界不等式中的有效绝对常数 $c_{LG}$，证明其至少可取值为 $2.0\times 10^{-17}$，该常数受 Maynard 多维筛法、算术级数素数分布及 Selberg 上界筛法系数等因素决定。
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