Related papers: Kloosterman sums in residue classes

We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi…

Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then…

We prove Sarnak's density conjecture for the principal congruence subgroup of SL_n(Z) of squarefree level and discuss various arithmetic applications. The ingredients include new bounds for local Whittaker functions and Kloosterman sums.

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…

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 revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to…

We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k=2, over general p-adic rings and even equal characteristic local rings. These can be…

In the previous paper [Sun23], the author proved a uniform bound for sums of half-integral weight Kloosterman sums. This bound was applied to prove an exact formula for partitions of rank modulo 3. That uniform estimate provides a more…

We obtain several asymptotic formulas for the sum of the divisor function $\tau(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced…

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and…

We give optimal bounds for matrix Kloosterman sums modulo prime powers extending earlier work of the first two authors on the case of prime moduli. These exponential sums arise in the theory of the horocyclic flow on $\mathrm{GL}_n$.

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the…

We study Kloosterman sums in a generalized ring-theoretic context, that of finite commutative Frobenius rings. We prove a number of identities for twisted Kloosterman sums, loosely clustered around moment computations.

We obtain lower bounds for the cardinality of $k$-fold sum-sets of reciprocals of elements of suitable defined short intervals in high degree extensions of finite fields. Combining our results with bounds for multilinear character sums we…

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 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…

In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in $\mathbb{Z}/m\mathbb{Z}$ for arbitrary integers $m$. These results have been motivated by a wide variety of…

In this paper, we study the power mean of a sum analogous to the Kloosterman sum by using analytic methods for character sums.

In this note, the polar decomposition of binary fields of even extension degree is used to reduce the evaluation of the Walsh transform of binomial Boolean functions to that of Gauss sums. In the case of extensions of degree four times an…

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…