Exponential Sums Research Articles

The character sum of polynomials with k variables and two-term exponential sums

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sums of polynomials with k variables and the two-term exponential sums, and give an identity and asymptotic formula for it.

Fast evaluation for the two-dimensional nonlinear coupled time–space fractional Klein–Gordon–Zakharov equations

In this paper, we develop a fast algorithm to solve the two-dimensional nonlinear coupled time–space fractional Klein–Gordon–Zakharov (KGZ) equations. The L2−1σ method based on an efficient sum-of-exponentials (SOE) approximation and a Fourier spectral method are used to approximate the time and space direction, respectively. And we use the previous time levels to deal with the nonlinear terms to obtain a linearized numerical scheme. Finally, a numerical example is given to show that our numerical method is of second order accuracy in time, spectral accuracy in space and the fast algorithm is effective.

Local densities of diagonal integral ternary quadratic forms at odd primes

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

Estimates for Weyl sums in the theory of discrete universality

We prove joint and disjoint discrete universality theorems for Dirichlet L-functions $$L(s,\chi )$$ and Hurwitz zeta-functions $$\zeta (s;\beta )$$ with rational parameter $$\beta \in (0,1]$$ . Our approach does not utilize Gallagher’s lemma, which is usually employed to prove discrete universality theorems. In our case, however, this would lead to certain difficulties when it is about estimating discrete second moments of L-functions. Therefore, we introduce a novel approach which is based only on Euler product representations and zero-density estimates of L-functions, as well as mean-value estimates for Weyl sums.

Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Let r n denote the number of representations of a positive integer n as a sum of two squares, i.e., n = x 1 2 + x 2 2 , where x 1 and x 2 are integers. We study the behavior of the exponential sum twisted by r n over the arithmetic progressions ∑ n ∼ X n ≡ l mod q r n e α n β , where 0 ≠ α ∈ ℝ , 0 < β < 1 , e x = e 2 π i x , and n ∼ X means X < n ≤ 2 X . Here, X > 1 is a large parameter, 1 ≤ l ≤ q are integers, and l , q = 1 . We obtain the upper bounds in different situations.

On the dynamical system generated by the Möbius transformation at prime times

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map \(x \mapsto (ax + b)/(cx+d)\) over a finite field of p elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.

A unified framework of SAGE and SONC polynomials and its duality theory

We introduce and study a cone which consists of a class of generalized polynomial functions and which provides a common framework for recent non-negativity certificates of polynomials in sparse settings. Specifically, this $\mathcal{S}$-cone generalizes and unifies sums of arithmetic-geometric mean exponentials (SAGE) and sums of non-negative circuit polynomials (SONC). We provide a comprehensive characterization of the dual cone of the $\mathcal{S}$-cone, which even for its specializations provides novel and projection-free descriptions. As applications of this result, we give an exact characterization of the extreme rays of the $\mathcal{S}$-cone and thus also of its specializations, and we provide a subclass of functions for which non-negativity coincides with membership in the $\mathcal{S}$-cone. Moreover, we derive from the duality theory an approximation result of non-negative univariate polynomials and show that a SONC analogue of Putinar's Positivstellensatz does not exist even in the univariate case.

Infinite Families of 2-Designs from a Class of Linear Codes Related to Dembowski-Ostrom Functions

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial [Formula: see text]-designs have attracted lots of research interest for decades. The interplay between coding theory and [Formula: see text]-designs started many years ago. It is generally known that [Formula: see text]-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a [Formula: see text]-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of [Formula: see text]-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of [Formula: see text]-designs are calculated explicitly.

A ratio of finitely many gamma functions and its properties with applications

In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of finitely many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, and the Bernstein function property, of the newly introduced ratio, and construct two inequalities of multinomial coefficients and multivariate beta functions.

The asymptotic number of zeros of exponential sums in critical strips

Normalized exponential sums are entire functions of the form $$\begin{aligned} f(z)=1+H_1e^{w_1z}+\cdots +H_ne^{w_nz}, \end{aligned}$$where \(H_1,\ldots , H_n\in {{\mathbb {C}}}\) and \(0<w_1<\cdots <w_n\). It is known that the zeros of such functions are in finitely many vertical strips S. The asymptotic number of the zeros in the union of all these strips was found by Langer (Bull Am Math Soc 37(4):213–239, 1931). Moreno (Compos Math 2(6):69–78, 1973) proved that there are zeros arbitrarily close to any vertical line in any strip S, provided that \(1,w_1,\ldots ,w_n\) are linearly independent over the rational numbers. In this study the asymptotic number of zeros in each individual vertical strip is found by relying on R. J. Backlund’s lemma, which was originally used to study the zeros of the Riemann \(\zeta \)-function. As a counterpart to Moreno’s result, it is shown that almost every vertical line meets at most finitely many small discs around the zeros of f.

More Constructions of 3-Weight Linear Codes

Linear codes with few weights have become an interesting research topic and important applications of cryptography and coding theory. In this paper, we apply some ternary near-bent and 2-plateaued functions or r -ary functions to construct more 3-weight linear codes, where r is a prime. Moreover, we determine the weight distributions of the resulted linear codes by means of some exponential sums.

Exponential Sums with Sparse Polynomials over Finite Fields

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 4 August 2020Accepted: 23 February 2021Published online: 11 May 2021Keywordsexponential sums, sparse polynomials, binomialsAMS Subject Headings11L07, 11T23Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Abstract Let $\varepsilon&gt; 0$ be sufficiently small and let $0 &lt; \eta &lt; 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

FAST COMPACT DIFFERENCE SCHEME FOR THE FOURTH-ORDER TIME MULTI-TERM FRACTIONAL SUB-DIFFUSION EQUATIONS WITH THE FIRST DIRICHLET BOUNDARY

<abstract> In this paper, a fast compact difference scheme is proposed for the initial-boundary value problem of fourth-order time multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions. Using the method of order reduction, the original problem can be converted to an equivalent lower-order system. Then at some super-convergence points, the multi-term Caputo derivatives are fast evaluated based on the sum-of-exponentials (SOE) approximation for the kernel functions appeared in Caputo fractional derivatives. The difficulty caused by the first Dirichlet boundary conditions is carefully handled. The energy method is used to illustrate the unconditional stability and convergence of the proposed fast compact scheme. The convergence accuracy is second-order in time and fourth-order in space if the solution has enough regularity. Compared with the direct scheme without the acceleration in time direction, the CPU time of the current fast scheme is largely reduced, which is shown by numerical examples. </abstract>

Estimates of short exponential sums with primes in major arcs

Estimates of short exponential sums with primes in major arcs

On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $ p $

The main purpose of this article is using the elementary methods and the properties of the character sums of the polynomials to study the calculating problem of one kind sixth power mean of the two-term exponential sums weighted by Legendre's symbol modulo $ p $, an odd prime, and give an interesting calculating formula for it.

On some additive problems of Goldbach’s type

In this paper, we find an asymptotic formula with power-saving remainder term for the number of solutions of one non-linear ternary problem with primes. The proof is based on the precise formulafor Chebyshev’s function involving the zeros of Riemann zeta function. In fact, a ternary problem each zerois solved. I. M. Vinogradov’s solution of the ternary Goldbach problem (1937, see [1], [2]) opened the way of solving a wide class of problems of the above type. In 1938, he found a power-saving estimate (with respect to the length of the summation interval) for the mean value of the modulus of the exponential sum with primes (see [2], theorem 3, p.82; theorems 6 and 7, p.86). Starting at 1996, G.I.Arkhipov, K.Buriev and the author have obtained several results concerning the exceptional sets in some binary problems of Goldbach’s type. These results used both the tools of the theory of Diophantine approximations and the precise formulasfrom Riemann’s zeta function theory. At the same time, the method of estimating of linear sums with primes based on the measure theory was derived in the papers of G. L. Watson, D. Bruedern, R. D. Cook and A. Perelli.

The remark on the mean value theorem for the absolute value of Dirichlet L-function in the critical strip

We continue our researches concerning the generalization and improvement of R.T.Turganaliev’s result that states an asymptotic formula for the mean value of the Riemann zeta function in the critical strip with power factor saving in the remainder term.We find an asymptotic for the mean value of Dirichlet L-function in the critical strip. This assertion improves R.T.Turganaliev’s theorem for zeta-function in the whole interval $$(1/2 < Re 𝑠 ≤ 1)$$. Our result is based on the special use of the estimation of exponential sums by second derivative test.

The hybrid power mean of some special character sums of polynomials and two-term exponential sums modulo $ p $

&lt;abstract&gt;&lt;p&gt;We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo $ p $, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them.&lt;/p&gt;&lt;/abstract&gt;

On the correlation of the sum of digits along prime numbers

Let $s_{q}$ denote the sum of digits function in base $q$. The aim of this work is to estimate the exponential sums involving the sum of digits of shifted prime numbers, of the form $\sum _{p\leq x}e(\alpha _{0}s_{q}(p)+\cdots +\alpha _{k}s_{q}(p+k))$, wh