Related papers: Kloosterman sums in residue classes

We establish cancellation in short sums of certain special trace functions over $\mathbb{F}_q[u]$ below the P\'{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the…

We return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's $\theta_\infty$-splitting function and a better choice of basis for…

Strong bounds - going beyond Sarnak's density hypothesis - are obtained for the number of automorphic forms for the congruence subgroup Gamma_0(q) of SL_n(Z) violating the Ramanujan conjecture at any given unramified place. The proof is…

We estimate the sums \[ \sum_{c\leq x} \frac{S(m,n,c,\chi)}{c}, \] where the $S(m,n,c,\chi)$ are Kloosterman sums of half-integral weight on the modular group. Our estimates are uniform in $m$, $n$, and $x$ in analogy with Sarnak and…

We obtain a nontrivial bound on the number of solutions to the equation $$ A^{x_1} + \ldots + A^{x_\nu} = A^{x_{\nu+1}} + \ldots + A^{x_{2\nu}}, \quad 1 \le x_1, \ldots,x_{2\nu} \le \tau, $$ with a fixed $n\times n$ matrix $A$ over a finite…

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for…

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by \'E. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, \'E. Fouvry, E. Kowalski, Ph. Michel and D. Mili\'cevi\'c (2017), E.…

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…

The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these…

We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential…

We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.

We formulate several analogues of the Chowla and Sarnak conjectures, which are widely known in the setting of the M\"obius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these…

We study Kloosterman sums on the orthogonal groups $SO_{3,3}$ and $SO_{4,2}$, associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums.…

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…

For a positive integer $m$ and a subgroup $\Lambda$ of the unit group $(\mathbb{Z}/m\mathbb{Z})^\times$, the corresponding generalized Kloosterman sum is the function $K(a,b,m,\Lambda) = \sum_{u \in \Lambda}e(\frac{au + bu^{-1}}{m})$.…

In this paper, we construct four binary linear codes closely connected with certain exponential sums over the finite field F_q and F_q-{0,1}. Here q is a power of two. Then we obtain four recursive formulas for the power moments of…

The maximal degree over rational numbers that an n-dimensinonal Kloosterman sum defined over a finite field of characteristic p can achieve is known to be (p-1)/d where d=gcd(p-1,n+1). Wan has shown that this maximal degree is always…

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 $GL(3)$…