Irrationality of the Angles of Kloosterman Sums over Finite Field

We prove that the angles of Kloosterman sums over arbitrary finite field are incommensurable with the constant π. In particular, this implies that the Weil bound for Kloosterman sums over finite fields cannot be reached.

Newton polygons for L-functions of generalized Kloosterman sums

In the present paper, we study the Newton polygons for the L-functions of n-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top-dimensional Dwork cohomology. Using Wan’s decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for the L-function of F ¯ ⁢ ( λ ¯ , x ) := ∑ i = 1 n x i a i + λ ¯ ⁢ ∏ i = 1 n x i - 1 , \bar{F}(\bar{\lambda},x):=\sum_{i=1}^{n}x_{i}^{a_{i}}+\bar{\lambda}\prod_{i=1}% ^{n}x_{i}^{-1}, with a 1 , … , a n {a_{1},\ldots,a_{n}} being pairwise coprime for n ≥ 2 {n\geq 2} .

Symplectic Kloosterman sums and Poincaré series

We prove power-saving bounds for general Kloosterman sums on $${\text {Sp}}(4)$$ Sp ( 4 ) associated to all Weyl elements via a stratification argument coupled with p-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of $${\text {Sp}}(4)$$ Sp ( 4 ) Poincare series.

On efficient computation of sums of characters on the basis of A. G. Postnikov methods

An efficient p-adic method and the structure of an algorithm for computing the sums of characters of finite abelian groups are presented. The method and algorithm are based on the A.G. Postnikov summation method of characters modulo a prime power and its developments. A brief survey of the theory of characters of finite abelian groups, p-adic arithmetic and analysis is presented. Questions of the efficiency of p-adic methods are discussed. Moreover, we present results of computation of other types of sums of characters (Kloosterman sums), which are connecting with Artin-Schreier coverings over prime finite fields. The corresponding method and algorithm are based on the development of another method by A.G. Postnikov. Examples of computation of sums of characters are given.

The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

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 able to construct families of $m$ -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.

On the size of the maximum of incomplete Kloosterman sums

Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$ . A classical problem in analytic number theory is bounding the maximum $$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$ of the absolute value of the incomplete sums $(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $ . In this very general context one of the most important results is the Pólya–Vinogradov bound $$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$ where $\hat t:{\mathbb F_p} \to \mathbb C$ is the normalized Fourier transform of t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any $\varepsilon > 0$ there exists a large subset of $a \in \mathbb F_p^ \times $ such that for $${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$ we have $$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$ as $p \to \infty $ . Finally, we prove a result on the growth of the moments of ${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$ .

A Hybrid Mean Value Involving Dedekind Sums and the Generalized Kloosterman Sums

In this paper, we use the mean value theorem of Dirichlet L -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.