Quadratic Character Research Articles

Quadratic characters with positive partial sums

AbstractLet be the set of all primes p for which the sums of over the interval [1, N] are non‐negative for all N. We prove that the estimate holds for .

Space-time duality in magnonics: Temporal diffraction in magnonic waveguides

In the present paper, we address the issue of the magnonic features due to the quadratic character of the dispersion of purely exchange SWs becoming more and more important due to a rapid progress in the technology of excitation of ultra-short SWs. Our interpretation in terms of time diffraction of the theoretical results and numerical simulations relies on the space/time duality principle that has already been successfully applied in ultrafast optics in the description of temporal evolution of short optical pulses. It was shown that time diffraction related effects lead to a considerable transformation of the signals transmitted through a magnonic channel in spintronic logic and computing devices that can be either detrimental or beneficial for the procedure of information processing. Specifically, time diffraction results in pulse expansion severely limiting the rate of information transfer. On the positive side, this feature can be effectively used to realize such important functionalities as pulse focusing/ compression and Fourier analysis which opens up novel promising opportunities in information processing.

Integral period relations and congruences

Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson-Flach element.

One-level density of the family of twists of an elliptic curve over function fields

We fix an elliptic curve E/Fq(t) and consider the family {E⊗χD} of E twisted by quadratic Dirichlet characters. The one-level density of their L-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (−1,1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ≠2 coprime to q. We obtain a restriction of (−1/2,1/2) on the support with a unitary symmetry.

Rédei permutations with the same cycle structure

Let Fq be the finite field of order q, and P1(Fq)=Fq∪{∞}. Write (x+y)m as N(x,y)+D(x,y)y. For m∈N and a∈Fq, the Rédei function Rm,a:P1(Fq)→P1(Fq) is defined by N(x,a)/D(x,a) if D(x,a)≠0 and x≠∞, and ∞, otherwise. In this paper we give a complete characterization of all pairs (m,n)∈N2 such that the Rédei permutations Rm,a and Rn,b have the same cycle structure when a and b have the same quadratic character and q is odd. We explore some relationships between such pairs (m,n), and provide explicit families of Rédei permutations with the same cycle structure. When a Rédei permutation has a unique cycle structure that is not shared by any other Rédei permutation, we call it isolated. We show that the only isolated Rédei permutations are the isolated Rédei involutions. Moreover, all our results can be transferred to bijections of the form mx and xm on certain domains.

Determination of GL(3) Cusp Forms by Central Values of Quadratic Twisted L-Functions

Abstract Let $\phi $ and $\phi ^{\prime}$ be two $\operatorname {GL}(3)$ Hecke–Maass cusp forms. In this paper, we prove that $\phi =\phi ^{\prime} \textrm {or }\widetilde {\phi ^{\prime}}$ if there exists a nonzero constant $\kappa $ such that $L(\frac {1}{2},\phi \otimes \chi _{8d})=\kappa L(\frac {1}{2},\phi ^{\prime}\otimes \chi _{8d})$ for all positive odd square-free positive $d$. Here, $\widetilde {\phi ^{\prime}}$ is dual form of $\phi ^{\prime}$ and $\chi _{8d}$ is the quadratic character $(\frac {8d}{\cdot })$. To prove this, we obtain asymptotic formulas for twisted 1st moment of central values of quadratic twisted $L$-functions on $\operatorname {GL}(3)$, which will have many other applications.

Steeping and dispersive effects analysis of a couple of long-wave equations in dispersive media

Dispersive wave propagations are described by nonlinear models, such as regularized long waves and modified regularized long waves, where nonlinearity and dispersion are important aspects of wave evolution to model long-wave propagation in dispersive media with small amplitudes. In this article, solitary wave solutions to the formerly indicated nonlinear equations are computed as exponential, hyperbolic, and trigonometric structures and their integration by balancing the steeping and dispersion terms of the highest order, and different classes of definitive localized coherent structures with their interaction properties, such as soliton solutions, are extracted by assigning appropriate functions. We have also investigated the impact of steeping and wave spread on waveforms. The wave solutions are assessed using the enhanced modified simple equation approach, which organizes the soliton solutions based on the features of the dispersive interconnections with the nonlinear cubic and quadratic characters that exist in the equations. The consistent 3D and 2D soliton configurations show that the wave profile is modulated by the wave velocity function and the parameters associated with it, and most of the modulation is due to linear effects.

Base change and triple product 𝐿-series

Let π i \pi _i be an irreducible cuspidal automorphic representation of G L 2 \mathrm {GL}_2 with central character ω i \omega _i . When ω 1 ω 2 ω 3 \omega _1\omega _2\omega _3 is trivial, Atsushi Ichino proved a formula for the central value L ( 1 2 , π 1 × π 2 × π 3 ) L(\frac {1}{2}, \pi _1\times \pi _2\times \pi _3) of the triple product L L -series in terms of global trilinear forms. We will extend this formula to the case when ω 1 ω 2 ω 3 \omega _1\omega _2\omega _3 is a quadratic character, giving a non-vanishing criterion of a local trilinear form in terms of the central value of the gamma factor.

On the Fourier coefficients of the Siegel Eisenstein series of odd level and the genus theta series

We give an explicit formula of the Siegel Eisenstein series of odd level, for any degree with trivial or quadratic character. For that we find the linear relation between the Siegel Eisenstein series of level p and the genus theta series, and show that the ramified Siegel series are given by a partial sum of the local densities.

Дослідження перебігу допалювання в порожнині робочого простору кисневого конвертера за рахунок оцінки фізичних параметрів димових газів та полум’я

The work is devoted to the development and evaluation of the effectiveness of an available method of monitoring the state of the converter bath based on the physical parameters of the flame formed during the afterburning of the source gases. In the process of purging in the oxygen converter, conditions are created for the oxidation reaction of CO to CO2 in the cavity of the unit. Control of the specified factor can allow corrective actions to be taken to ensure a high level of technological and environmental indicators and be an indicator of the completeness of the flow of chemical transformations in the cavity of the oxygen converter. according to the luminosity of the gas torch in different spectrum; by the electrical conductivity of the gas torch. The conditions of different conditions of the high-temperature environment are taken into account: dustiness; fractions of dust emitted from the converter and their chemical composition, completeness of combustion when different fractions of the oxidizer are supplied. Simulation of the state of the environment in the upper part of the oxygen converter was performed. It was established that the most influential negative factor in the assessment of afterburning is dust, which affects all physical parameters of the flame and at high temperatures of the process itself becomes an object of radiation of both heat and light. According to the results of the research, it was concluded that it is most appropriate to measure the electrical conductivity of the flame between the torch and the blowing nozzle in order to have more complete information about the afterburning processes. When evaluating the operation of the blowing device, it was established that the dependence of the electrical conductivity of the flame on the CO content can serve as a characteristic of afterburning indicators and has a typical quadratic character.

One Kind New Hybrid Power Mean and Its Computational Formulae

The main purpose of this study is to use the elementary and analytic methods and the properties of the classical Gauss sums to study the calculation problems of one kind of hybrid power mean involving the quadratic character sums and the two‐term exponential sums and give an exact computational formula for it.

Non-radiative relaxation and nonlinear properties of YVO4:Yb3+, Er3+ upconversion nanoparticles

• Analysis of actual energy transfer channels in YVO 4 :Yb 3+ ,Er 3+ UCNP is presented. • Simple (reduced) model for energy transfer processes is proposed. • Pump power dependence of upconversion photoluminescence is yielded. • It is established that multiphonon transitions mainly effect on upconversion nonlinearity. • Theoretical results could be useful in UCNPs characterization by spectroscopic data analysis. We analyzed nonlinear emission properties of oxide nanocrystals YVO 4 doped with Yb 3+ and Er 3+ ions which are one of the most interesting types of upconverting nanoparticles (UCNPs). As is widely believed the deviation of the power-dependent upconversion phosphorescence intensity from a quadratic character is attributed to a saturation effect occurring at the relatively high pump power. In this paper, we focused on studying upconversion nonlinearity at the low pump power when excited states populations are considerably less then ground states populations. The analysis of actual energy transfer channels allowed us to propose a reduced model having simple solutions. The obtained analytic expressions for the dependences of upconversion photoluminescence intensity and its nonlinearity on the pump power revealed that multiphonon transitions are a main factor responsible for the variation of upconversion nonlinearity in a range between 1 and 2. This result could be used in spectroscopic experiments for simple and direct identification of the non-radiative relaxation efficiency in YVO 4 :Yb, Er UCNPs.

Multiple Lucas–Dirichlet Series Associated With Additive and Dirichlet Characters

In this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas L-function associated with Dirichlet characters and additive characters. We then compute a list of possible singularities and residues of these functions at these poles. Further, we show the rationality of the multiple Lucas L-functions associated with quadratic characters at negative integer arguments.

A Bombieri–Vinogradov Theorem for Higher-Rank Groups

Abstract We establish a result of Bombieri–Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n$ over a number field $F$ when $\pi$ is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau–Siegel zeros. In particular, when $\pi $ is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted $L$-values $|L(1,\pi \otimes \chi )|$ in the $\chi $-aspect, where $\chi $ is a primitive quadratic Hecke character over $F$. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a $\textrm {GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\textrm {GL}_n\times \textrm {GL}_2$ shifted convolution sum.

Quadratic torsion subgroups of modular Jacobian varieties

Let D be an odd square-free positive integer and C be a divisor of D. For any quadratic character x modulo C, we prove that a large portion of the x-part of the torsion subgroup of J0(DC) coincides with that of its cuspidal subgroup.

Weighted Measures of Pseudorandom Binary Lattices

In cryptography one needs pseudorandom sequences whose short subsequences are also pseudorandom. To handle this problem, Dartyge, Gyarmati and Sárközy introduced weighted measures of pseudorandomness of binary sequences. In this paper we continue the research in this direction. We introduce weighted pseudorandom measure for multidimensional binary lattices and estimate weighted pseudorandom measure for truly random binary lattices. We also give lower bounds for weighted measures of even order and present an example by using the quadratic character of finite fields.

Dirichlet $L$-functions of quadratic characters of prime conductor at the central point

We prove that more than nine percent of the central values L(1/2,\chi_p) are non-zero, where p\equiv 1 \pmod{8} ranges over primes and \chi_p is the real primitive Dirichlet character of conductor p . Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of L(1/2,\chi_p) , and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.

Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

A short note on inadmissible coefficients of weight 2 and 2k+1 newforms

Let f(z)=q+sum _{nge 2}a(n)q^n be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for k=2 by ruling out or locating all odd prime values |ell |<100 of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights kge 1 newforms where the nebentypus is given by a quadratic Dirichlet character.

Certain transformations and special values of hypergeometric functions over finite fields

In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we evaluate certain character sums to establish these transformations. We then use these transformations to evaluate explicitly several special values of $${_2}F_1$$ hypergeometric functions over finite fields. Certain special values of $${_2}F_1$$ hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from our special values of $${_2}F_1$$ hypergeometric functions containing arbitrary characters as parameters. One of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu also follows from one of our transformation formulas.