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。
摘要：
我们改进了所有 k≥4 情形下带有 k 次幂模的大筛法不等式。我们的方法将这些不等式与华林问题的一个受限变体联系起来。首先，我们引入一个经典的除数界，该界描述正整数表示为两个 k 次幂之和的表示数。其次，我们引入 Wooley 最近关于指数和均值的一般性结果。最后，我们陈述一个条件性结果，该结果基于猜想性的 Hardy-Littlewood 公式，该公式描述大正整数表示为 k+1 个 k 次幂之和的表示数。
英文摘要：
We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k$th-powers. Secondly, we input a recent and general result of Wooley on mean values of exponential sums. Lastly, we state a conditional result, based on the conjectural Hardy-Littlewood formula for the number of representations of a large positive integer as a sum of $k+1$ $k$th-powers.
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.
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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.
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.
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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…
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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--…
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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.
Sign changes of Kloosterman sums with moduli having at most six prime factors
School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）；Research Center for Number Theory and Its Applications, Northwest University（西北大学数论及其应用研究中心）
We show that if an essentially arbitrary sequence supported on an interval containing $x$ integers, is convolved with a tiny Siegel-Walfisz-type sequence supported on an interval containing $\exp((\log x)^{\varepsilon})$ integers then the resulting multiplicative convolution has (in a weak sense) level of distribution $x^{1/2 + 1/66 - \varepsilon}$ as $x$ goes to infinity. This dispersion estimate has a number of consequences for: the distribution of the $k$th divisor function to moduli $x^{1/2 + 1/66 - \varepsilon}$ for any integer $k \geq 1$, the distribution of products of exactly two prime…
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…
We investigate three combinatorial problems considered by Erdös, Rivat, Sarközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.
