Related papers: Kloosterman sums with twice-differentiable functio…

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 this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon.…

For $m,n>0$ and $mn<0$ we estimate the sums \begin{equation*} \sum_{c \leq x} \frac{S(m,n,c,\chi)}{c}, \end{equation*} where the $S(m,n,c,\chi)$ are Kloosterman sums attached to a multiplier $\chi$ of weight $1/2$ on the full modular group.…

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…

In this paper we study incidences for hyperbolas in $\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of…

The series of some new estimates for the sums of the type \[ S_{q}(x;f)\,=\,\mathop{{\sum}'}\limits_{n\leqslant x}f(n)e_{q}(an^{*}+bn) \] is obtained. Here $q$ is a sufficiently large integer, $\sqrt{q}(\log{q})\!\ll\!x\leqslant q$, $a,b$…

The study of $n$-dimensional inverted Kloosterman sums was suggested by Katz (1995) who handled the case when $n=1$ from complex point of view. For general $n\geq 1$, the $n$-dimensional inverted Kloosterman sums were studied from both…

We give a construction of radial Fourier interpolation formulas in dimensions 3 and 4 using Maass--Poincar\'e type series. As a corollary we obtain explicit formulas for the basis functions of these interpolation formulas in terms of what…

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 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…

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 obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include…

Let $\F_q$ ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in $\F_q[x]$ with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is…

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are…

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…

In this note, we deduce an asymptotic formula for even power moments of Kloosterman sums based on the important work of N. M. Katz on Kloosterman sheaves. In a similar manner, we can also obtain an upper bound for odd power moments.…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

In a recent work by Charpin, Helleseth, and Zinoviev Kloosterman sums $K(a)$ over a finite field $\F_{2^m}$ were evaluated modulo 24 in the case $m$ odd, and the number of those $a$ giving the same value for $K(a)$ modulo 24 was given. In…

We obtain statistical results on the possible distribution of all partial sums of a Kloosterman sum modulo a prime, by computing explicitly the support of the limiting random Fourier series of our earlier functional limit theorem for…

Let $\mathcal{K}(a)$ denote the Kloosterman sum on the finite field of order $2^n$. We give a simple characterization of $\mathcal{K}(a)$ modulo 16, in terms of the trace of $a$ and one other function. We also give a characterization of…