In this paper, we construct three ternary linear codes associated with the orthogonal group O^-(2,q) and the special orthogonal groups SO^-(2,q) and SO^-(4,q). Here q is a power of three. Then we obtain recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of "Gauss sums" for the orthogonal and special orthogonal groups O^-(2n,q) and SO^-(2n,q).
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).
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.
Exotic and inverted Kloosterman sums over semisimple algebras
Center for Discrete Mathematics, Chongqing University（重庆大学离散数学中心）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉科学研究院）
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.
By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from $\ell$-adic cohomology, as well as a Bombieri--Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke--Maass cusp form of trivial nebentypus, the eigenvalue of the $n$-th Hecke operator does not coincide with the Kloosterman sum $\mathrm{Kl}(1,n)$ for infinitely many squarefree $n$ with at most $100$ prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterm…
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.
Some applications of smooth bilinear forms with Kloosterman sums
我们重新审视了最近的一个约束。Shparlinski和T. P. Zhang关于具有Kloosterman和的双线性型的结果，并证明了Kloosterman和对模形式的傅里叶系数的相关和的一个推广.我们使用这些界限，以改善早期的结果Kloosterman和沿素数和Dirichlet $L$ -函数的四阶矩的误差项。
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the above character sum as the character runs over all non-trivial multiplicative characters mod $p.$ Moreover, an arcsine law is also established as a consequence of the method of moments. The evaluations of these moments also allow us to obtain asymptotic formulae for moments of such character sums weighted by special $L$-values (at $1/2$ and $1$).
We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum $\sum_{p\leq x} \Kl(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\Kl(1, n)$, where $n$ runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\left| \Kl(1, n)\right|$.
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.
