Kloosterman sums do not correlate with periodic functions
Raphael S. Steiner
本文为所有等差数列中的Kloosterman和之和提供了一致界，并由此证明Kloosterman和与周期函数不相关。
我们为所有等差数列中的Kloosterman和之和提供了一致界。由此得出，Kloosterman和与周期函数不相关。
We provide uniform bounds for sums of Kloosterman sums in all arithmetic progressions. As a consequence, we find that Kloosterman sums do not correlate with periodic functions.
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Sums of Kloosterman sums over square-free and smooth integers
本文受近期关于 Kloosterman 和的双线性型及扭曲和的研究启发，研究了由无平方因子整数和光滑整数参数化的 Kloosterman 和之和的若干新问题。
We use some elementary arguments to obtain a new bound on bilinear sums with weighted Kloosterman sums which complements those recently obtained by E. Kowalski, P. Michel and W. Sawin (2020).
We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.
We obtain several estimates for trilinear form with double Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums.
We prove two results on Kloosterman sums over finite fields, using Stickelberger's theorem and the Gross-Koblitz formula. The first result concerns the minimal polynomial over Q of a Kloosterman sum, and the second result gives a characterisation of ternary Kloosterman sums modulo 27.
The main purpose of this paper is to study the hybrid mean value problem involving generalized Dedekind sums, generalized Hardy sums and Kloosterman sums, and give some exact computational formulae for them by using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function.
We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $Γ_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum functi…
Let $c_q(n)$ be the Ramanujan sums and let $S(m,n;q)$ be the Kloosterman sums. We study the eigenvalues of $q \times q$ matrices whose $(m,n)$ entry is $c_q (m-n)$ or $S(m,n;q)$ where $q$ is a fixed positive integer. We also study the eigenvalues of matrices whose entries are sums of Ramanujan sums or sums of Kloosterman sums.
We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.
We give results characterising ternary Kloosterman sums modulo 9 and 27. This leads to a complete characterisation of values that ternary Kloosterman sums assume modulo 18 and 54. The proofs uses Stickelberger's theorem, the Gross-Koblitz formula and Fourier analysis.
An elementary proof of the Selberg identity for Kloosterman sums
本文利用加性特征的正交性，给出了Kloosterman和的Selberg恒等式的初等证明。
We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.
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We show that sums of the SL(3,Z) long element Kloosterman sum against a smooth weight function have cancellation due to the variation in argument of the Kloosterman sums, when each modulus is at least the square root of the other. Our main tool is Li's generalization of the Kuznetsov formula on SL(3,R), which has to date been prohibitively difficult to apply. We first obtain analytic expressions for the weight functions on the Kloosterman sum side by converting them to Mellin-Barnes integral form. This allows us to relax the conditions on the test function and to produce a partial inversion fo…
We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.
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The equidistribution of Elliptic Dedekind sums and generalized Selberg-Kloosterman sums
该研究证明归一化椭圆戴德金和模1等分布，核心是用庞加莱级数给出PSL₂(ℂ)离散子群的广义Selberg-Kloosterman和的非平凡界，推进了数论中相关和的分布研究。
Exotic and inverted Kloosterman sums over semisimple algebras
本文引入非交换有限维半单代数上的奇异Kloosterman和与奇异倒置Kloosterman和，证明其可约化为交换étale代数上的相应和，并给出平方根估计。
A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of $GL(2)$ automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using $GL(3)$ automorphic forms, resolving a long-standing problem of Bump, Friedberg and Goldfeld. Along the way, we develop what are apparently new bounds for the order derivatives of the classical $J$-Bessel function, and we conclude with a discussion of the original method of Bump, Friedberg and Goldfeld.
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Sums of Kloosterman sums formed with modular symbols
研究由模符号形成的克洛斯特曼和的和，利用陶伯型方法给出相关拉马努金和的和的估计，定义类似塞尔伯格zeta函数的函数并确定其解析延拓，证明和的抵消结论，解释与特征值问题联系并给出数值证据。
We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.
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Distinctness of the generalized $n$-dimensional Kloosterman sums
通过伽罗瓦理论和p进高斯和的施蒂克尔贝格尔定理，研究有限域\(F_q\)上一类广义n维克洛斯特曼和的代数次数与差异性。
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Properties of Kloosterman sums for long Weyl element on GL(n)
本文针对GL(n)长Weyl元的Kloosterman和，提出一种基于Cauchy-Binet公式、无需Plücker坐标的新表达式，并据此证明其重要性质。
We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel, and Milićević (2014) and Fouvry, Kowalski and Michel (2014).
This paper has been withdrawn by the author because Proposition 1 is not valid.
We derive explicit formulas for some Kloosterman sums on $Γ_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.
This paper has been withdrawn by the author because there is a gap in Lemma 9.
In the present paper, we generalize some of the results on Kloosterman sums proven in \cite{BG} for prime moduli to general moduli. This requires to establish the corresponding additive properties of the reciprocal set $$ I^{-1}=\{x^{-1}:\quad x\in I\}, $$ where $I$ is an interval in the ring of residue classes modulo a large positive integer. We apply our bounds on multilinear exponential sums to the Brun-Titchmarsh theorem and the estimate of very short Kloosterman sums, hence generalizing our earlier work to the setting of general modulus.
We compute the second moment of spinor $L$-functions at central points of Siegel modular forms on congruence subgroups of large prime level $N$ and give applications to non-vanishing.
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.
For a positive integer $m$ and a subgroup $Λ$ of the unit group $(\mathbb{Z}/m\mathbb{Z})^\times$, the corresponding generalized Kloosterman sum is the function $K(a,b,m,Λ) = \sum_{u \in Λ}e(\frac{au + bu^{-1}}{m})$. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.
Triple sums of Kloosterman sums and the discrepancy of modular inverses
研究大区间内模正整数$c$的模逆分布，给出其盒、球及各向同性偏差的上下界以展示与随机点集的偏离，分析基于克洛斯特曼和三重和的新界。
The Weil bound for generalized Kloosterman sums of half-integral weight
本文证明了附着于偶格的Weil表示的半整数权重广义Kloosterman和的Weil界，并通过证明一个将除数和与稀疏指数和关联的恒等式，推广了Kohnen的恒等式。
Symplectic Kloosterman Sums and Poincaré Series
本文通过分层论证和 p-adic 驻相方法，证明了 Sp(4) 上一般 Kloosterman 和的幂节省界，并将其与 Poincaré 级数的 Fourier 系数相关联。
For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where $\text{Kl}_2(p;q)$ denotes a normalised Kloosterman sum with modulus $q$. This is a sparse analogue of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Milićević showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimisation argument inspired by Fouvry, Kowalski and Michel. Our argument compares three different bo…
On algebraic degrees of inverted Kloosterman sums
本文研究了反转 $n$ 维 Kloosterman 和作为代数整数的代数次数，扩展了 Katz 及作者前作中从复数与 $p$-adic 视角对该和的研究。
Symplectic Kloosterman Sums for $\operatorname{Sp}(2n)$ with Powerful Moduli
本文研究了Sp(2n)的辛Kloosterman和的界，并将其应用于共 prime 对称对的等分布。
We construct p-adic relative cohomology for a family of toric exponential sums which generalize the classical Kloosterman sums. Under natural hypotheses such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except in the top dimension. Our construction enables sufficiently sharp estimates for the action of Frobenius on cohomology so that our earlier work may be applied to the L-functions coming from linear algebra operations on these families to deduce a number of basic properties.
The classical $n$-variable Kloosterman sums over finite fields are well understood by Deligne's theorem from complex point of view and by Sperber's theorem from $p$-adic point of view. In this paper, we study the complex and $p$-adic estimates of inverted $n$-variable Kloosterman sums, addressing a question of N. Katz (1995). We shall give two complex estimates. The first one is elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological results of Adolphson-Sperber, Denef-Loeser and Fu for twisted toric exponential sums. This deeper result assumes that the ch…
Kloosterman sums on orthogonal groups
本文研究 $SO_{3,3}$ 与 $SO_{4,2}$ 上关联 Weyl 群短元素的 Kloosterman 和，用多维指数和给出显式描述，并结合代数几何与 $p$-进分析建立界。
Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article considers the case of short sums of normalized classic…
For a prime $p$, we consider Kloosterman sums $$ K_{p}(a) = \sum_{x\in \F_p^*} \exp(2 πi (x + ax^{-1})/p), \qquad a \in \F_p^*, $$ over a finite field of $p$ elements. It is well known that due to results of Deligne, Katz and Sarnak, the distribution of the sums $K_{p}(a)$ when $a$ runs through $\F_p^*$ is in accordance with the Sato--Tate conjecture. Here we show that the same holds where $a$ runs through the sums $a = u+v$ for $u \in \cU$, $v \in \cV$ for any two sufficiently large sets $\cU, \cV \subseteq \F_p^*$. We also improve a recent bound on the nonlinearity of a Boolean function asso…
We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel and Milićević and also of Fouvry, Kowalski and Michel. In particular, we improve the bound on the error term in the asymptotic formula for mixed moments of $L$-series associated with Hecke eigenforms.
Matrix Kloosterman sums modulo prime powers
本文研究素数幂模的矩阵 Kloosterman 和，将既有素数模结果推广到素数幂情形，给出最优界，并指出其与 $\mathrm{GL}_n$ 上 horocyclic 流的联系。
Sums of products of Kloosterman sums to prime power moduli
本文针对素幂模下的Kloosterman和乘积和，证明了新的强幂节省界，通过利用p进可微性的初始项估计完全指数和的方法实现，为相关数论问题提供了量化的对齐分析。
Matrix Kloosterman sums
本文研究 $\mathrm{GL}_n$ 上环面流中产生的一族指数和，证明了一般纯性的显式版本，并得到了这些和的最优界。
Bounds for Kloosterman sums on GL(n)
本文通过建立指数和的显式表示，获得了 GL(n) 上 Kloosterman 和的幂次节省界，并以此超越素数级主同余子群的 Sarnak 密度猜想，同时界定了 GL(4) 的所有 Kloosterman 和。
Kloosterman sums for a finite field arise as Frobenius trace functions of certain local systems defined over $\Gm$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic Galois representation that comes from geometry. We also give bounds on the ramification of these Galois representations. All this is done in the g…
We construct a binary linear code $C(O(3,q))$, associated with the orthogonal group $O(3,q)$. Here $q$ is a power of two. Then we obtain a recursive formula for the odd power moments of Kloosterman sums with trace one arguments in terms of the frequencies of weights in the codes $C(O(3,q))$ and $C(Sp(2,q))$. This is done via Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups.
In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are…
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