Gauss Sums Research Articles

Generalized quadratic Gauss sums and their 2mth power mean

Abstract The main purpose of this article is to study the problem of calculating the 2mth power mean of the generalized quadratic Gauss sums, and using the analytic method and an interesting combinatorial identity to give a sharp asymptotic formula for the 2mth power mean. Thus, a new simple proof of this existing result [N. Bag, A. Rojas-León, and W. P. Zhang, On some conjectures on generalized quadratic Gauss sums and related problems, Finite Fields Appl. 86 (2023), 10213] is provided.

Bias in cubic Gauss sums: Patterson's conjecture

Bias in cubic Gauss sums: Patterson's conjecture

One special kind of Kloosterman sum and its fourth-power mean

Abstract This article aims to investigate the calculation problem of the fourth-power mean of the specific Kloosterman sums by utilizing analytic methods and the properties of classical Gauss sums. It provides a captivating calculation formula for this purpose. Additionally, an effective method for studying high-power mean of Kloosterman sums is introduced in this research.

Prime number factorization and degree of coherence of speckled light beams.

We discover a connection between a Gauss sum of number theory and the degree of coherence (DOC) of the field in a transverse plane of structured speckled light beams. We theoretically demonstrate and experimentally validate that prime number factorization can be achieved by manipulating the source beam's DOC in Young's double-slit experiment. The determination of whether a number can be factored is based solely on the visibility of the resulting interference patterns. Our findings offer new insights into information encryption and decryption, data compression, etc.

Theoretical, structural, and electronic analyses of pyridin-based dyes for dye-sensitized solar cells applications.

The new series of donor-π-acceptor dyes have been designed using pyridine derivatives as a donor group and thienothiophene as a π-spacer group, which were linked via 10acceptor groups. The highest occupied molecular orbital energies range from - 6.177 to - 5.786eV, whereas the lowest unoccupied molecular orbital energies range from - 2.181 to - 3.664eV. A6 dye has smaller energy gap, lower hardness, higher electrophilicity index, and good photovoltaic performance than other sensitizers. The lowest dihedral angle is observed in A1, A2, A6, A7, and A8 which are appropriate for intramolecular charge transfer between the molecules. The A8 has higher light harvesting efficiency, which increases the photovoltaic efficiency of the designed dye. The A6, A7, and A8 dyes spend less time in the excited state, which means they emit photons more efficiently than other dyes. The interaction between donor to π-spacer (red line) parts of the dyes has the bonding interaction (positive), and π-spacer to acceptor (blue line) parts of the dyes have the bonding and antibonding (negative) behaviours. The dyes A5 and A9 have 305.79 and 357.71 times higher β0 values than urea (0.781 × 10-30 esu) molecules. The spectral properties of the A6 dye strongly affect the structural modification. The density functional theory (DFT) and time-dependent DFT (TD-DFT) approach B3LYP/6-311G (d,p) basic set were used to optimize the designed dyes. All the calculations are performed using Gauss view 6.0 and Gaussian 09 software. The density of state spectrum is plotted using Gauss sum 2.6.

Unveiling the potential of graphene and S-doped graphene nanostructures for toxic gas sensing and solar sensitizer cell devices: insights from DFT calculations.

In this work, we explore the potential of 2D materials, particularly graphene and its derivatives, for eco-friendly electricity generation and air pollution reduction. Leveraging the significant surface area of graphene nanomaterials, the susceptibility of these graphene-based nanostructures to hazardous substances and their applicability in clean solar cell (SSC) devices were systematically investigated using density functional theory (DFT), as implemented within Gaussian 5.0 code. Time-dependent DFT (TD-DFT) was employed to characterize the UV-visible spectrum of unstrained nanostructures. Herein, we considered three potentially harmful gases-CO, NH3, and Br2. Adsorption calculations revealed a notable interaction between the pure graphene nanostructure and Br2 gas, while the S-doped counterpart exhibited reduced interaction. Saturated S-doped nanostructures demonstrated an enhanced affinity for NH3 and CO gases compared to their pure S-doped counterparts, attributed to the sulfur (S) atom facilitating gas molecule binding to the nanostructure's surface. Furthermore, simulations of the SSC device architecture indicated the superior performance of the pure graphene nanostructure in terms of light-harvesting efficiency, injection energy, and electron injection into the lower conduction band of CBM titanium dioxide (TiO2). These findings suggest a potential avenue for developing nanostructures tailored for SSC devices and gas sensors, offering a dual solution to address air pollution concerns. Density function theory was used to compute the ground and excited state properties for pure and sulfur-doped graphene nanostructures. The hybrid function B3LYP with a 6-31G* basis set was utilized to describe the exchange correlation. Gauss Sum 2.2 software is used to estimate the density of state (DOS) for all structures under investigation.

Beyond Gauss Sum in “Proof by Tomatoes”

An insightful way of understanding how the sum of cubes from 13 to n3 can be represented by the square of the Gauss sum from 1 to n.

A matrix variant of the Erdős-Falconer distance problems over finite field

We study a matrix analog of the Erdős-Falconer distance problems in vector spaces over finite fields. We provide lower bounds for the cardinality of a subset of the matrix algebra over a finite field such that its distance set is the whole matrix algebra. There arises an interesting analysis of certain quadratic matrix Gauss sums.

Some New Identities Related to Dedekind Sums Modulo a Prime

The main purpose of this article is to use some identities of the classical Gauss sums, the properties of character sums, and Dedekind sums (modulo an odd prime) to study the computational problem of one-kind mean values related to Dedekind sums and give some interesting identities for them.

On the generalized exponential sums and their fourth power mean

Abstract The main purpose of this article is to study the calculating problem of the fourth power mean of the two-term exponential sums and provide an accurate calculating formula for utilizing analytical methods and character sums’ properties. In the meantime, a result of the fourth power mean of Gauss sums is improved.

Vanishing of quartic and sextic twists of L-functions

Let E be an elliptic curve over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}. We conjecture asymptotic estimates for the number of vanishings of L(E,1,χ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L(E,1,\\chi )$$\\end{document} as χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document} varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on E. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} twisted elliptic curve L-functions, with ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} even and composite, is a mixed family with both unitary and orthogonal aspects.

On the Fourier coefficients of Siegel Eisenstein series of degree 3 of an odd prime level with the quadratic character

In this paper we give an explicit formula of the Fourier coefficients of the Siegel Eisenstein series of degree 3, level $p$ with quadratic character. For the calculation we use the result of the generalized Gauss sum computed by Hiroshi Saito. After the tedious calculation we can get a rather simple result.

Bilinear forms with Kloosterman and Gauss sums in function fields

In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in Z/mZ for arbitrary integers m. These results have been motivated by a wide variety of applications, such as improved asymptotic formulas for moments of L-functions. However, there has been very little work done in this area in the setting of rational function fields over finite fields. We remedy this and provide a number of new non-trivial bounds for bilinear forms of Kloosterman and Gauss sums in this setting, based on new bounds on the number of solutions to certain modular congruences in Fq[T]. These improve upon some results of Macourt and Shparlinski (2019).

The quotient set of the quadratic distance set over finite fields

Abstract Let 𝔽 q d {\mathbb{F}_{q}^{d}} be the d-dimensional vector space over the finite field 𝔽 q {\mathbb{F}_{q}} with q elements. For each non-zero r in 𝔽 q {\mathbb{F}_{q}} and E ⊂ 𝔽 q d {E\subset\mathbb{F}_{q}^{d}} , we define W ⁢ ( r ) {W(r)} as the number of quadruples ( x , y , z , w ) ∈ E 4 {(x,y,z,w)\in E^{4}} such that Q ⁢ ( x - y ) Q ⁢ ( z - w ) = r {\frac{Q(x-y)}{Q(z-w)}=r} , where Q is a non-degenerate quadratic form in d variables over 𝔽 q {\mathbb{F}_{q}} . When Q ⁢ ( α ) = ∑ i = 1 d α i 2 {Q(\alpha)=\sum_{i=1}^{d}\alpha_{i}^{2}} with α = ( α 1 , … , α d ) ∈ 𝔽 q d {\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{F}_{q}^{d}} , Pham (2022) recently used the machinery of group actions and proved that if E ⊂ 𝔽 q 2 {E\subset\mathbb{F}_{q}^{2}} with q ≡ 3 ⁢ ( mod ⁡ 4 ) {q\equiv 3~{}(\operatorname{mod}\,4)} and | E | ≥ C ⁢ q {|E|\geq Cq} , then we have W ⁢ ( r ) ≥ c ⁢ | E | 4 q {W(r)\geq\frac{c|E|^{4}}{q}} for any non-zero square number r ∈ 𝔽 q {r\in\mathbb{F}_{q}} , where C is a sufficiently large constant, c is some number between 0 and 1, and | E | {|E|} denotes the cardinality of the set E. In this article, we improve and extend Pham’s result in two dimensions to arbitrary dimensions with general non-degenerate quadratic distances. As a corollary of our results, we also generalize the sharp results on the Falconer-type problem for the quotient set of distance set due to the first two authors and Parshall (2019). Furthermore, we provide improved constants for the size conditions of the underlying sets. The key new ingredient is to relate the estimate of the W ⁢ ( r ) {W(r)} to a quadratic homogeneous variety in 2 ⁢ d {2d} -dimensional vector space. This approach is fruitful because it allows us to take advantage of Gauss sums which are more handleable than the Kloosterman sums appearing in the standard distance-type problems.

The hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums

<abstract><p>This article applied the properties of character sums, quadratic character, and classical Gauss sums to study the calculations of the hybrid power mean of the generalized Gauss sums and the generalized two-term exponential sums. It also provided exact formulas for calculating these hybrid power means.</p></abstract>

Boundedness of Gaussian Random Sums on Rooted Homogeneous Trees

Boundedness of Gaussian Random Sums on Rooted Homogeneous Trees

The fourth power mean of the generalized quadratic Gauss sums associated with some Dirichlet characters

<abstract><p>In this paper, the fourth power mean values of the generalized quadratic Gauss sums associated with the $ 3 $-order and $ 4 $-order Dirichlet characters are given by using the properties of the Dirichlet characters and Gauss sums.</p></abstract>

Bounds on the norms of maximal operators on Weyl sums

We obtain new estimates on the maximal operator applied to the Weyl sums, which are tight in some cases. We also consider the quadratic case (that is, Gauss sums) in more details. In wide ranges of parameters our estimates are optimal and match lower bounds. Our approach is based on a combination of ideas of Baker (2021) and Chen and Shparlinski (2020).

Prospective utilization of boron nitride and beryllium oxide nanotubes for Na, Li, and K-ion batteries: a DFT-based analysis.

In the present work, we investigated the adsorption mechanism of natural sodium (Na), potassium (K), and lithium (Li) atoms and their respective ion on two nanostructures: boron-nitride nanotubes (BNNTs) and beryllium-oxide nanotubes (BeONTs). The main goal of this research is to calculate the gain voltage for Na, K, and Li ionic batteries. Density function theory (DFT) calculations indicated that the adsorption energy between Na + is higher than that of the other cations, and this is particularly clear in the BeONT. Furthermore, gain voltage calculations showed that BNNTs generate a higher potential than BeONTs, with the most significant difference observed in BNNT/Na + . This research provides theoretical insights into the potential uses of these nanostructures as anodes in Na, K, and Li-ion batteries. Density function theory used to compute the ground state properties for BeONT and BNNT with and without selected atoms and their ions (Li, K, and Na). B3LYP used for exchange correlation between electrons and ions, and 6-31G* basis set used for all atoms and ions. Gauss Sum 2.2 software used for estimate the density of state (DOS) for all structure under investigation.

Dirichlet Series with Periodic Coefficients, Riemann’s Functional Equation, and Real Zeros of Dirichlet L-Functions

ABSTRACT In this paper, we provide Dirichlet series with periodic coefficients that have Riemann’s functional equation and real zeros of Dirichlet L-functions. The details are as follows. Let L(s, χ) be the Dirichlet L-function and G(χ) be the Gauss sum associated with a primitive Dirichlet character χ (mod q). We define f ( s , χ ) : = q s L ( s , χ ) + i − κ ( χ ) G ( χ ) L ( s , χ ¯ ) , where χ ¯ is the complex conjugate of χ and κ(χ) := (1 – χ(−1))/2. Then, we prove that f (s, χ) satisfies Riemann’s functional equation in Hamburger’s theorem if χ is even. In addition, we show that f (σ, χ) ≠ 0 for all σ ≥ 1. Moreover, we prove that f(σ, χ) ≠ 0 for all 1/2 ≤ σ < 1 if and only if L(σ, χ) ≠ 0 for all 1/2 ≤ σ < 1. When χ is real, all zeros of f (s, χ) with ℜ ( s ) > 0 are on the line σ = 1/2 if and only if the generalized Riemann hypothesis for L(s, χ) is true. However, f (s, χ) has infinitely many zeros off the critical line σ = 1/2 if χ is non-real.