Related papers: Visual properties of generalized Kloosterman sums
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…
In this paper power saving bounds for general Kloosterman sums for all Weyl elements for $\mathrm{GL}_n$ for $n>2$ are proven, improving the trivial bound by D\k{a}browski and Reeder. This is achieved by representing the sums in an explicit…
We prove that the Kloosterman sum $\text{Kl}(1,q)$ changes sign infinitely many times, as $q\rightarrow +\infty$ with at most six prime factors. As a consequence, our result improved the best known result of Xi(IMRN, 2022). The novelty of…
Mutual-visibility sets were motivated by visibility in distributed systems and social networks, and intertwine with several classical mathematical areas. Monotone properties of the variety of mutual-visibility sets, and restrictions of such…
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…
Lie symmetries of K(m,n) equations with time-dependent coefficients are classified. Group classification is presented up to widest possible equivalence groups, the usual equivalence group of the whole class for the general case and…
Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic…
We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb…
We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…
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 generalize the universal power series of Seleznev to several variables and we allow the coefficients to depend on parameters. Then, the approximable functions may depend on the same parameters. The universal approximation holds on…
In this paper we give an essential treatment for estimating arbitrary integral power moments of Kloosterman sums over the residue class ring. For prime moduli we derive explicit estimates, and for prime-power moduli we prove concrete…
We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any…
For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…
In this paper, we generalize the Bohr-Rogosinski sum for the Ma-Minda classes of starlike and convex functions. Also the phenomenon is studied for the classes of starlike functions with respect to symmetric points and conjugate points along…
We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form $\sum_{k=0}^n a_k x_{n,k}$ for given sequences of vectors $(x_{n,k})_{n\geq k\geq 0}$ in a topological vector…
We give new bounds for $\sum_{{a, m ,n}}\alpha_{m}\beta_n\nu_a {\textrm e}\left(\frac{a\overline m}{n}\right)$ where $\alpha_{m}$, $\beta_n$ and $\nu_a$ are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec…
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.…
In this paper, we obtain a new class of $p$-ary binomial bent functions which are determined by Kloosterman sums. The bentness of another three classes of functions is characterized by some exponential sums and some results in…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.