Kloosterman Sums Research Articles

One special kind of Kloosterman sum and its fourth-power mean

Abstract This article aims to investigate the calculation problem of the fourth-power mean of the specific Kloosterman sums by utilizing analytic methods and the properties of classical Gauss sums. It provides a captivating calculation formula for this purpose. Additionally, an effective method for studying high-power mean of Kloosterman sums is introduced in this research.

Sums of multidimensional Kloosterman sums

AbstractWe obtain a new bound on certain multiple sums with multidimensional Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums in very small families.

The Weil bound for generalized Kloosterman sums of half-integral weight

Abstract Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

Random walks and the “Euclidean” association scheme in finite vector spaces

Abstract In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $ ) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

Uniform bounds for Kloosterman sums of half-integral weight with applications

Abstract Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to x with implied constants depending on m and n. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in x, m and n. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of x with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in x, m and n. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to m and n, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function p ⁢ ( n ) {p(n)} . We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo 3, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

Degrees of generalized Kloosterman sums

Abstract The modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers.

Bilinear forms with Kloosterman and Gauss sums in function fields

In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in Z/mZ for arbitrary integers m. These results have been motivated by a wide variety of applications, such as improved asymptotic formulas for moments of L-functions. However, there has been very little work done in this area in the setting of rational function fields over finite fields. We remedy this and provide a number of new non-trivial bounds for bilinear forms of Kloosterman and Gauss sums in this setting, based on new bounds on the number of solutions to certain modular congruences in Fq[T]. These improve upon some results of Macourt and Shparlinski (2019).

Fourth power mean of the general s-dimensional Kloosterman sum mod p

In this article, we prove an asymptotic formula for the fourth power mean of a general s-dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod p which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem.

Bessel functions and Kloosterman integrals on GL(n)

This paper will focus on the proof of local integrability of Bessel functions for GL(n) (p-adic case) by using the relations between Bessel functions and local Kloosterman (orbital) integrals proved in several papers of E. M. Baruch [5][6][7], the theory of the (relative) Shalika germs established by H. Jacquet and Y. Ye in [25][26] and G. Stevens' approach [35] on estimating certain GL(n) generalized Kloosterman sums.

Kloosterman sums do not correlate with periodic functions

AbstractWe 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.

On the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum

On the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum

Generalization of the Lehmer problem over incomplete intervals

Let alpha geq 2, mgeq 2 be integers, p be an odd prime with pnmid m (m+1 ), 0<lambda _{1} , lambda _{2}leq 1 be real numbers, q=p^{alpha }> max { [ frac{1}{lambda _{1}} ], [ frac{1}{lambda _{2}} ] }. For any integer n with (n,q)=1 and a nonnegative integer k, we define Mλ1,λ2(m,n,k;q)=∑′a=1q∑′b=1[λ1q]∑′c=1[λ2q]ab≡1(modq)c≡am(modq)n∤b+c(b−c)2k.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ M_{\\lambda _{1},\\lambda _{2}} ( m,n,k;q )=\\mathop{\\mathop{ \\mathop{\\mathop{{\\sum }'}_{a=1}^{q}\\mathop{{\\sum }'}_{b=1}^{ [ \\lambda _{1}q ]}\\mathop{{\\sum }'}_{c=1}^{ [\\lambda _{2}q ]}}_{ab\\equiv 1(\\bmod q)}}_{c\\equiv a^{m}(\\bmod q)}}_{n\ mid b+c} ( b-c )^{2k}. $$\\end{document} In this paper, we study the arithmetic properties of these generalized Kloosterman sums and give an upper bound estimation for it. By using the upper bound estimation, we discuss the properties of M_{lambda _{1},lambda _{2}} ( m,n,k;q ) and obtain an asymptotic formula.

On values of the Bessel function for generic representations of finite general linear groups

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of GLn(Fq). We show that special values of the Bessel function can be realized as the coefficients of L-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.

Newton polygons of L-functions for two-variable generalized Kloosterman sums

In this paper, we study the Newton polygon of the [Formula: see text]-function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the [Formula: see text]-adic complex associated to the [Formula: see text]-function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary.

Chowla and Sarnak conjectures for Kloosterman sums

AbstractWe formulate several analogs of the Chowla and Sarnak conjectures, which are widely known in the setting of the Möbius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.

On sums of hyper-Kloosterman sums

Abstract A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of GL ⁡ ( 2 ) {\operatorname{GL}(2)} automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using GL ⁡ ( 3 ) {\operatorname{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.

On the mean value of the generalized Dedekind sum and certain generalized Hardy sums weighted by the Kloosterman sum

UDC 511 We study a hybrid mean-value problem related to the generalized Dedekind sum, certain generalized Hardy sums, and Kloosterman sum and obtain several meaningful conclusions by means of the analytic method and the properties of the character sum and the Gauss sum.

Moments of mixed Kloosterman sums, supercharacters, and elliptic curves

Moments of mixed Kloosterman sums, supercharacters, and elliptic curves

Distribution of Kloosterman paths to high prime power moduli

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p n p^n of a fixed odd prime p p , a pure depth-aspect analogue of theorems of Kowalski–Sawin and Ricotta–Royer–Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as n → ∞ n\to \infty to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every p p converges in law as p → ∞ p\to \infty , and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the p p -adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments.

Bounds on bilinear forms with Kloosterman sums

AbstractWe prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet ‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.