Fixed Complex Number Research Articles

Note on Type L-Functions of Euler Product of Second Degree

In this paper we are concerned with a special case of Euler functions. We shall study the Euler product of degree two (Type of L-Euler functions) and give some results. More precisely, we shall deal with some Dirichlet series associated with a class of arithmetic {an} under the condition that apapk = apk+1 + pαapk−1, provided p is prime, k≥1, and α is a fixed complex number. We will demonstrate that there is an Euler’s product for the Dirichlet series Σnan/ns. This result is important in analysis, especially in analytic number theory.

Riemann-type functional equations: Julia line and counting formulae

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau’s formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.

Joint Value-Distribution of Shifts of the Riemann Zeta-Function

We prove that any complex values z_1,ldots ,z_n can be approximated by the following integral shifts of the Riemann zeta-function zeta (s+id_1tau ),ldots ,zeta (s+id_ntau ) for infinitely many tau , provided d_1,ldots ,d_nin {mathbb {Q}} are distinct and positive, and s is a fixed complex number lying in the right open half of the critical strip.

Duhamel Banach algebra structure of some space and related topics

Let ? be a fixed complex number, and let ? be a simply connected region in complex plane C that is starlike with respect to ? ? ?. We define some Banach space of analytic functions on ? and prove that it is a Banach algebra with respect to the ?-Duhamel product defined by (f?? g)(z) := d/dz z?? f(z+??t)g(t)dt. We prove that its maximal ideal space consists of the homomorphism h? defined by h?(f)=f(?). Further, we characterize the lattice of invariant subspaces of the integration operator J?f(z)=?z? f(t)dt. Moreover, we describe in terms of ?-Duhamel operators the extended eigenvectors of J?.

On the archimedean and nonarchimedean 𝑞-Gevrey orders

q q -Difference equations appear in various contexts in mathematics and physics. The “basis” q q is sometimes a parameter, sometimes a fixed complex number. In both cases, one classically associates to any series solution of such equations its q q -Gevrey order expressing the growth rate of its coefficients : a (nonarchimedean) q − 1 q^{-1} -adic q q -Gevrey order when q q is a parameter, an archimedean q q -Gevrey order when q q is a fixed complex number. The objective of this paper is to relate these two q q -Gevrey orders, which may seem unrelated at first glance as they express growth rates with respect to two very different norms. More precisely, let f ( q , z ) ∈ C ( q ) [ [ z ] ] f(q,z) \in \mathbb {C}(q)[[z]] be a series solution of a linear q q -difference equation, where q q is a parameter, and assume that f ( q , z ) f(q,z) can be specialized at some q = q 0 ∈ C × q=q_{0} \in \mathbb {C}^{\times } of complex norm > 1 >1 . On the one hand, the series f ( q , z ) f(q,z) has a certain q − 1 q^{-1} -adic q q -Gevrey order s q s_{q} . On the other hand, the series f ( q 0 , z ) f(q_{0},z) has a certain archimedean q 0 q_{0} -Gevrey order s q 0 s_{q_{0}} . We prove that s q 0 ≤ s q s_{q_{0}} \leq s_{q} “for most q 0 q_{0} ”. In particular, this shows that if f ( q , z ) f(q,z) has a nonzero (nonarchimedean) q − 1 q^{-1} -adic radius of convergence, then f ( q 0 , z ) f(q_{0},z) has a nonzero archimedean radius converges “for most q 0 q_{0} ”.

Associated orthogonal polynomials of the first kind and Darboux transformations

Let u be a quasi-definite linear functional defined on the linear space of polynomials P with complex coefficients. For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) (Pn)n∈N, which satisfies a three term recurrence relation. If we increase the indices of the recurrence relation by one unity, then we get the sequence of associated polynomials of the first kind as solution. These polynomials will be orthogonal with respect to a linear functional denoted by u(1). In the literature two special transformations of the functional u are studied, the canonical Christoffel transformation u˜=(x−c)u and the canonical Geronimus transformation uˆ=(x−c)−1u+Mδc, where c is a fixed complex number, M is a free complex parameter and δc is the linear functional defined on P as 〈δc,p(x)〉=p(c). For the Christoffel transformation with SMOP (P˜n)n∈N, we are interested in analyzing the relation between the linear functionals u(1) and u˜(1). The super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind (Pn(1))n∈N and (P˜n(1))n∈N, respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations. To get this result, we first need to study the relation between u−1 (the inverse functional) and u(1) which can be given from a quadratic Geronimus transformation.

Analytic and Combinatorial Number Theory

Класс рядов Дирихле, соответствующих периодическим арифметическим функциям $f$, включает в себя дзета-функцию Римана и $L$-функции характеров Дирихле. В работе изучается распределение значений таких рядов Дирихле $L(s;f)$ и их аналитических продолжений в окрестности критической прямой (которая является осью симметрии соответствующего функционального уравнения типа Римана). В частности, для заданного комплексного числа $a\neq 0$ и четных или нечетных периодических функций $f$ находится количество $a$-значений $\Delta $-множителя функционального уравнения, доказывается существование среднего для значений рядов Дирихле $L(s;f)$ в этих точках, показывается, что ординаты этих $a$-значений равномерно распределены по модулю $1$, и этот результат применяется для доказательства дискретной теоремы универсальности.

Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line

The class of Dirichlet series associated with a periodic arithmetical function $$f$$ includes the Riemann zeta-function as well as Dirichlet $$L$$ -functions to residue class characters. We study the value-distribution of these Dirichlet series $$L(s;f)$$ and their analytic continuation in the neighbourhood of the critical line (which is the axis of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $$a\neq 0$$ , we find for an even or odd periodic $$f$$ the number of $$a$$ -points of the $$\Delta$$ -factor of the functional equation, prove the existence of the mean of the values of $$L(s;f)$$ taken at these points, show that the ordinates of these $$a$$ -points are uniformly distributed modulo one and apply this to show a discrete universality theorem.

On Asymptotics of Solutions to Some Linear Differential Equations

In this paper, we find the principal asymptotic term at infinity of a certain fundamental system of solutions to the equation l2n[y] = λy of order 2n, where l2n is the product of second-order linear differential expressions and λ is a fixed complex number. We assume that the coefficients of these differential expressions are not necessarily smooth but have a prescribed power growth at infinity. The asymptotic formulas obtained are applied for the problem on the defect index of differential operators in the case where l2n is a symmetric (formally self-adjoint) differential expression.

Integrable representations for toroidal extended affine Lie algebras

Let g be any untwisted affine Kac–Moody algebra, μ any fixed complex number, and g˜(μ) the corresponding toroidal extended affine Lie algebra of nullity two. For any k-tuple λ=(λ1,⋯,λk) of weights of g, and k-tuple a=(a1,⋯,ak) of distinct non-zero complex numbers, we construct a class of modules V˜(λ,a) for the extended affine Lie algebra g˜(μ). We prove that the g˜(μ)-module V˜(λ,a) is completely reducible. We also prove that the g˜(μ)-module V˜(λ,a) is integrable when all weights λi in λ are dominant. Thus, we obtain a new class of irreducible integrable weight modules for the toroidal extended affine Lie algebra g˜(μ).

On the Stability of First Order PDEs Associated to Vector Fields

Let \(\mathcal {M}\) be a manifold, \(\mathcal {V}\) be a vector field on \(\mathcal {M}\), and \(\mathcal {B}\) be a Banach space. For any fixed function \(f:\mathcal {M}\rightarrow \mathcal {B}\) and any fixed complex number \(\lambda \), we study Hyers–Ulam stability of the global differential equation \(\mathcal {V}y=\lambda y+f\).

Additive ρ-functional inequalities in β-homogeneous normed spaces

In this paper, we solve the following additive ?-functional inequalities ||f (x + y) - f (x) - f (y)|| ? ???(2f (x+y/2) - f(x) + -f (y))??, (1) where ? is a fixed complex number with |?|<1, and ??2f(x+y/2)-f(x)- f(y)???||?(f(x+y)-f(x)-f(y))||, (2) where ? is a fixed complex number with |?|<1/2 , and prove the Hyers-Ulam stability of the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive ?-functional equations associated with the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces.

Benjamini–Schramm continuity of root moments of graph polynomials

Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini–Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc.In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v0 the measures arising from the Tutte polynomial ZGn(z,v0) converge in holomorphic moments if the sequence (Gn) of finite graphs is Benjamini–Schramm convergent. This answers a question of Abért and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler.

Quadratic $\rho$-functional inequalities and equations

In this paper, we solve the quadratic $\rho$-functional inequalities where $\rho$ is a fixed complex number with $|\rho| where $\rho$ is a fixed complex number with $|\rho|< \frac{1}{2}$. Furthermore, we prove the Hyers-Ulam stability of the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic $\rho$-functional equations associated with the quadratic $\rho$-functional inequalities (0.1) and (0.2) in complex Banach spaces.

On the occurrence of Hecke eigenvalues and a lacunarity question of Serre

Let \pi be a unitary cuspidal automorphic representation for GL(n) over a number field. We establish upper bounds on the number of Hecke eigenvalues of \pi equal to a fixed complex number. For GL(2), we also determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number \gamma; in the case \gamma=0, this answers a question of Serre. These bounds are then improved upon by restricting to non-dihedral representations. Finally, we obtain analogous bounds for a family of cuspidal automorphic representations for GL(3).

ZEROS OF THE ESTERMANN ZETA FUNCTION

AbstractIn this paper we investigate the zeros of the Estermann zeta function $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable $s$, where $k$ and $\ell $ are coprime integers and ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number $\alpha $. In particular, we study the question on how the zeros of $E(s; k/ \ell , \alpha )$ depend on the parameters $k/ \ell $ and $\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

A result concerning two-sided centralizers on algebras with involution

The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L (X) be the algebra of all bounded linear operators on X and let A (X)⊂ L (X) be a standard operator algebra, which is closed under the adjoint operation. Let T : A (X)→L (X) be a linear mapping satisfying the relation 3T (AA∗A) = T (A)A∗A+AT(A∗)A+AA∗T (A) for all A ∈ A (X). In this case T is of the form T (A) = λA for all A ∈ A (X), where λ is some fixed complex number. Mathematics subject classification (2010): 16N60, 46B99, 39B42.

Asymptotics of the colored Jones function of a knot

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the $n$th colored Jones polynomial at $e^{\a/n}$, when $\a$ is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1/n$ when $\a$ is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the $n$th colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when $\a$ is near $2 \pi i$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.

THE MAXIMAL VALUE OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS

Let be a fixed complex number. In this paper, we study the quantity , where is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given and , the quantity S(, n) can be calculated. Then we compute the limit for every of modulus 1. It is equal to 1/ if is not a root of unity. If , where and k [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin(/d)) and 1/(2d sin(/2d)) for d = 1, d even and d > 1 odd, respectively.

Quotients of values of the Dedekind Eta function

Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form $${\frac{\eta(A_jz)} {\eta(A_{j-1}z)}}$$ , where A j-1 and A j are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in $${\mathbb{R}^2}$$ , and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.