Related papers: Explicit Evaluation of Double Gauss Sums

Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\gcd(N,p)=1$, and let $\k$ be a primitive multiplicative character of order $N$ over finite field $\fq$. This paper studies the problem…

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…

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…

Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$ S=\sum_{n\in \mathcal{N}}\left|\sum_{m\in \mathcal{M} }e_p(an g^{m})\right|, \quad \gcd…

We determine explicitly the Gauss sums on the general linear group $GL_2(\mathbb{Z}/p^l\mathbb{Z})$ for all irreducible characters, where $p$ is an odd prime and $l$ is an integer > 1. While there are several studies of the Gauss sums on…

We show that for any mod $p^m$ characters, $\chi_1, \dots, \chi_k,$ the Jacobi sum, $$ \sum_{x_1=1}^{p^m}\dots \sum_{\substack{x_k=1\\x_1+\dots+x_k=B}}^{p^m}\chi_1(x_1)\dots \chi_k(x_k), $$ has a simple evaluation when $m$ is sufficiently…

The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived.

Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…

Let $R$ be a finite ring with unity, $\psi: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( \chi_0 \) be the principal multiplicative character ($i.e.$, $\chi_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss…

An asymptotic expansion for the generalised quadratic Gauss sum $$S_N(x,\theta)=\sum_{j=1}^{N} \exp (\pi ixj^2+2\pi ij\theta),$$ where $x$, $\theta$ are real and $N$ is a positive integer, is obtained as $x\rightarrow 0$ and…

Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum…

In this paper we study the sum of powers in the Gaussian integers $\mathbf{G}_k(n):=\sum_{a,b \in [1,n]} (a+b i)^k$. We give an explicit formula for $\mathbf{G}_k(n) \pmod n $ in terms of the prime numbers $p \equiv 3 \pmod 4$ with $p \mid…

Let $N$ be a positive number. We give an asymptotic formula for the sum of $\tau(\gcd(a,b))$ for all $a$ and $b$ with $ab \le N$.

In the past two decades, many researchers have studied {\it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$ for the…

In this paper, we will explicitly calculate Gauss sums for the general linear groups and the special linear groups over $\Bbb Z_n$, where $\Bbb Z_n=\Bbb Z/n \Bbb Z$ and $n>0$ is an integer. For $r$ being a positive integer, the formulae of…

We derive a formula for the evaluation of weighted generalized Fibonacci sums of the type $S_k^n (w,r) = \sum_{j = 0}^k {w^j j^r G_j{}^n }$. Several explicit evaluations are presented as examples.

The Gauss circle problem concerns with the evaluation of $\sum_{n \leq x}r(n)$, where $r(n)$ denotes the number of representations of $n$ as sums of two squares and $x \geq 2$. Let $\Psi_G(x,y)$ denote the sum of $y$-smooth numbers below…

For the Gauss sums which are defined by S_n(a,q) := \sum_{x (mod q)} e(ax^n/q), Stechkin (1975) conjectured that the quantity A := \sup_{n,q\ge 2} \max_{\gcd(a,q)=1} |S_n(a,q)|/q^(1-1/n) is finite. Shparlinski (1991) proved that A is…

For a natural number $m$, generalized $m$-gonal numbers are defined by the formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum…

A multiple Gauss sum is a complete multiple exponential sum twisted by Dirichlet characters. We prove a new bound for multiple Gauss sums and, as an application, improve previous results in the Birch--Goldbach problem. Let $F_1, \ldots, F_R…