Related papers: Distribution of differences of characters evaluate…

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which…

We provide estimates for sums of the form \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\chi(a+b+c)\right|\] and \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}\chi(a+b+cd)\right|\] when $A,B,C,D\subset \mathbb F_p$, the field…

Let $E$ be an elliptic curve over the finite field $\mathbb{F}_p$, and $P \in E(\mathbb{F}_p)$ be an $\mathbb{F}_p$-rational point. We study the sums \[ S_{\chi,P}(N,h) = \sum_{n=1}^N \chi(\psi_n(P)) \chi(\psi_{n+h}(P)), \] where…

In this paper, we investigate the distribution of values of the complete exponential sum $S_{p,\chi}(\theta)=\sum_{n=1}^p \chi(n)e(n\theta)$, where $p$ is a large prime, $\chi$ is a Dirichlet character (mod $p$) of order $d\geq 2$, and…

We estimate double sums $$ S_\chi(a, I, G) = \sum_{x \in I} \sum_{\lambda \in G} \chi(x + a\lambda), \qquad 1\le a < p-1, $$ with a multiplicative character $\chi$ modulo $p$ where $I= \{1,\ldots, H\}$ and $G$ is a subgroup of order $T$ of…

Let $p$ be an odd prime. Using I. M. Vinogradov's bilinear estimate, we present an elementary approach to estimate nontrivially the character sum $$ \sum_{x\in H}\chi(x+a),\qquad a\in\Bbb F_p^*, $$ where $H<\Bbb F_p^*$ is a multiplicative…

We obtain explicit estimates for the mixed character sum $S= S(\chi,g,f,p^m) = \sum_{x=1}^{p^m} \chi (g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $\chi$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over…

We estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the…

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…

Let $E$ be an elliptic curve over the finite field $\mathbb{F}_p$, and $P \in E(\mathbb{F}_p)$ be an $\mathbb{F}_p$-rational point. We obtain nontrivial estimates for multiplicative character sums associated with the division polynomials…

In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski.…

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, ...$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of…

Let $M(\chi)$ denote the maximum of $|\sum_{n\le N}\chi(n)|$ for a given non-principal Dirichlet character $\chi \pmod q$, and let $N_\chi$ denote a point at which the maximum is attained. In this article we study the distribution of…

We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical…

We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} \chi (x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} \chi_1 (x)\chi_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for…

For a large prime $p$, and a polynomial $f$ over a finite field $F_p$ of $p$ elements, we obtain a lower bound on the size of the multiplicative subgroup of $F_p^*$ containing $H\ge 1$ consecutive values $f(x)$, $x = u+1, \ldots, u+H$,…

For a prime number $p$ and integer $x$ with $\gcd(x,p)=1$ let $\overline{x}$ denote the multiplicative inverse of $x$ modulo $p.$ In the present paper we are interested in the problem of distribution modulo $p$ of the sequence $$…

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le…

Let $p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as…