Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_1,\dots,χ_m)$ ($χ_1\dotsm χ_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $χ_1\in \mathcal{A}_1,\dots, χ_m\in \mathcal{A}_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$, if $\#\mathcal{A}_1\ge q^{\frac{1}{2}+ε}$, $\#\mathcal{A}_2 \ge (\log q)^{\frac{1}δ-1}$ for $ε>δ>0$ fixed and $q\to \infty$ or if $\#\mathcal{A}_1\#\mathcal{A}_2/q\to \infty$. This extends previous results of Xi, Z. Zheng, and the authors.
Jacobi sums and cyclotomic numbers are the important objects in number theory. The determination of all the Jacobi sums and cyclotomic numbers of order $e$ are merely intricate to compute. This paper presents the lesser numbers of Jacobi sums and cyclotomic numbers which are enough for the determination of all Jacobi sums and the cyclotomic numbers of a particular order.
The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain algebraic and arithmetic characterizations for Jacobi sums of order 2l^2 .
The aim of this paper is to deal with congruences for Jacobi sums of order $2l^{2}$ over a finite field $\mathbb{F}_{q}, q=p^{r}$, $p^{r}\equiv 1\ (mod \ 2l^{2})$, where $l>3$ and $p$ are primes. Further, we also calculate Jacobi sums $J_{2l^{2}}(1,n)$ of order $2l^{2}$ in terms of Dickson-Hurwitz sums.
The determination of Jacobi sums, their congruences and cyclotomic numbers have been the object of attention for many years and there are large number of interesting results related to these in the literature. This survey aims at reviewing results concerning the diophantine systems for finding the cyclotomic numbers and coefficients of Jacobi sums and to indicate the current status of the problem.
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's $q$-trigonometric identities.
Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $χ: \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(χ^{\ell}, χ^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.
We show that for any mod $p^m$ characters, $χ_1, \dots, χ_k,$ the Jacobi sum, $$ \sum_{x_1=1}^{p^m}\dots \sum_{\substack{x_k=1\\x_1+\dots+x_k=B}}^{p^m}χ_1(x_1)\dots χ_k(x_k), $$ has a simple evaluation when $m$ is sufficiently large (for $m\geq 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\geq 2$.
We prove an estimate for multi-variable multiplicative character sums over affine subspaces of $\mathbb A^n_k$, which generalize the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.
Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $J(χ,η)/\sqrt{q}$, for each fixed non-trivial multiplicative character $η$, becomes equidistributed in the unit circle as $q\rightarrow+\infty,$ when $χ$ runs over all non-trivial multiplicative characters different from $η^{-1}.$ Previously, the similar equidistribution was obtained by Katz and Zheng by varying both of $χ$ and $η$. On the other hand, we also obtain the equidistribution of $J(χ,η)$ as $(χ,η)$ runs over $\mathcal{X}\times\mathcal{Y}\subseteq(\widehat{\mathrm{F}^*})^2$, as long as $|\mat…
In this paper, we study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in \cite{feng1}, where the Jacobi sums over Galois rings with characteristics a square of a prime number were discussed.
Let us consider a cyclic extension of a function field defined over a finite field. For a character (non-trivial) of this extension, we calculate, as a linear combinations of products of Jacobi sums, the coefficients of the polynomial given by its Dirichtlet $L$-series. In the last section we show applications of this calculation.
Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $χ_1,\dots, χ_m$ of $\mathbf{F}_q^\times$, we let $J(χ_1,\dots, χ_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(χ_1,χ_2)$ ($χ_1χ_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $χ_1$ and $χ_2$ run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$. In this paper, we show a similar property for $m\ge 2$. More generally, we show that the normalized Jacobi sum $q^{-(m-1)/2}J(χ_…
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…
We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and $E$. We apply our method to determine generalizations of a family of $\text{sech}$-sums given by Ramanujan and generalizations of a family of $\text{csch}$-sums given by Zucker. Our method has the advantage of producing evaluations for hyperbolic sums with sign functions that have not previously appeared in the literature on hyperbolic sums. We apply our met…
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 $\sum f(x_1,\ldots,x_k)$, where the summation is taken over all pairwise distinct $x_1,\ldots,x_k \in G$. In particular, let $p^s$ be a power of an odd prime, $n$ a positive integer coprime with $p-1$, and $a_1,\ldots,a_k$ integers such that $φ(p^s)$ divides $a_1+\cdots+a_k$ and $p-1$ does not divide $\sum_{i \in I}a_i$ for all non-empty proper subsets $I\subseteq \{1,\…
We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of $M$-ary Sidelnikov sequences over $\mathbb{F}_q$, where $q$ is a power of an odd prime $p$. We show that the cross-correlation values of pairs of sequences from such a family can be expressed in terms of certain Jacobi sums. This insight facilitates the computation of the cross-correlation values of these sequence pairs so long as $ϕ(M)^{ϕ(M)} \leq q.$ We are also able to use our Jacobi sum expression to deduce explicit formulae for the cross-correlation distr…
In the first part, we consider generalized quadratic Gauss sums as finite analogues of the Jacobi theta function, and the reciprocity law for Gauss sums as their transformation formula. We attach finite Dirichlet series to Gauss sums using a Möbius transform, and show they have a functional equation, Euler product factorization, and roots only on the critical line. In the second part, we prove a general reciprocity law for sums of exponentials of rational quadratic forms in any number of variables, using the transformation formula of the Riemann theta function on the Siegel upper half-space.
We study the problem of tiling and packing in vector spaces over finite fields, its connections with zeroes of classical exponential sums, and with the Jacobian conjecture
Binary Sidel'nikov-Lempel-Cohn-Eastman sequences (or SLCE sequences) over F 2 have even period and almost perfect autocorrelation. However, the evaluation of the linear complexity of these sequences is really difficult. In this paper, we continue the study of [1]. We first express the multiple roots of character polynomials of SLCE sequences into certain kinds of Jacobi sums. Then by making use of Gauss sums and Jacobi sums in the "semiprimitive" case, we derive new divisibility results for SLCE sequences.
We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.
