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Two Results Related to the Universality of Zeta-Functions with Periodic Coefficients

Roma Kačinskaitė, Brigita Kazlauskaitė

Results in Mathematics > 2018 > 73 > 3 > 1-19

In the paper, we give two results on the investigation of mixed joint universality property of the zeta-functions with periodic coefficients, i.e., we study the joint functional independence and denseness for the collections consisting of periodic zeta- and periodic Hurwitz zeta-functions. We show some generalizations as well.

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Ewald expansions of a class of zeta-functions

Kalyan Chakraborty, Shigeru Kanemitsu, Haruo Tsukada

SpringerPlus > 2016 > 5 > 1 > 1-16

The incomplete gamma function expansion for the perturbed Epstein zeta function is known as Ewald expansion. In this paper we state a special case of the main formula in Kanemitsu and Tsukada (Contributions to the theory of zeta-functions: the modular relation supremacy. World Scientific, Singapore, 2014) whose specifications will give Ewald expansions in the H-function hierarchy. An Ewald expansion...

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Zeta-invariants of the Steklov spectrum of a planar domain

E. G. Mal’kovich, V. A. Sharafutdinov

Siberian Mathematical Journal > 2015 > 56 > 4 > 678-698

The classical inverse problem of the determination of a smooth simply-connected planar domain by its Steklov spectrum [1] is equivalent to the problem of the reconstruction, up to conformal equivalence, a positive function $$a \in C^\infty (\mathbb{S})$$ on the unit circle $$(\mathbb{S}) = \left\{ {e^{i\theta } } \right\}$$ from the spectrum of the operator aΛ_e, where Λ_e = (−d ²/dθ ²...

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Universality of Various Zeta-Functions

Roma Kačinskaitė

Electronic Notes in Discrete Mathematics > 2013 > 43 > Complete > 129-135

This is a survey of results on joint universality in Voroninʼs sense of various zeta-functions, when in the collection of these functions some of them have the Euler product and the others have not.

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The parameterized-Euler-constant function γα(z)

Li-meng Xia

Journal of Number Theory > 2013 > 133 > 1 > 1-11

We define parameterized-Euler-constant function z↦γα(z):=∑n=1∞zn−1(αn−logn+αn) with parameter α>−1, defined on the closed unit disk. The function γα(z) is an extension of the function z↦γ(z):=γ1(z) studied recently and having the property that γ1(1)=C, C being the Euler–Mascheroni constant. Integrals for γα(z) provide its analytic continuation to C∖[1,+∞).

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On the instability of the Riemann hypothesis for curves over finite fields

P.M. Gauthier, N. Tarkhanov

Journal of Approximation Theory > 2012 > 164 > 4 > 504-515

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) an analog of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. No number theory is required...

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On the instability of the Riemann Hypothesis for varieties over finite fields

F. Donzelli, P. M. Gauthier

Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) > 2012 > 47 > 3 > 124-133

There exist perturbations of a rational function which remove zeroes and poles from a prescribed region as well as perturbations which add zeroes and poles to a prescribed region. We employ this to show the instability of the Riemann Hypothesis for zeta-functions of smooth projective varieties over finite fields.

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Some Four Dimensional Definite Integrals Arising from Zeta-Function Theory

Richard R. Hall

Computational Methods and Function Theory > 2012 > 12 > 2 > 565-583

The integral $$\mathop \int\int\int\limits_\Delta\int (\alpha + \beta)^{k} (\gamma + \delta)^{l}(\alpha + \gamma)^{m}(\beta+\delta)^{n}\ d\alpha \ d\beta \ d\gamma \ d\delta$$ in which $\Delta=\lbrace (\alpha,\beta,\gamma,\delta)\in [0,1]^{4}:\alpha + \beta + \gamma + \delta \leq 1\rbrace$ occurs in a formula of J. B. Conrey. We evaluate this integral in closed form in some special cases...

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On the exceptional fields for a class of real quadratic fields

Li Liu, Hongwen Lu

Acta Mathematica Scientia > 2011 > 31 > 3 > 1179-1188

In this paper, we give a lower bound exp(2.2 × 10 ⁸ ) for those discriminants of real quadratic fields Q ( d ) with d = N ² − 4 and h(d) = 1.

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On münts formula

Ding Xiaqi, Luo Peizhu

Acta Mathematica Scientia > 2010 > 30 > 1 > 132-136

In the theory of Riemann Zeta-function, there is an important formula called Münts formula. In this note we will give its general form.

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Random and algebraic permutations’ statistics

Vladimir I. Arnold

Functional Analysis and Other Mathematics > 2009 > 2 > 2-4 > 247-248

An algebraic permutation $\hat{A}\in S(N=n^{m})$ is the permutation of the N points of the finite torus ℤ _n ^m , realized by a linear operator A∈SL(m,ℤ_n). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements,...

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On Min's zeta-function and Hermite elliptic operator

Xiaqi Ding, Yi Ding

Acta Mathematica Scientia > 2007 > 27 > 3 > 449-455

In this note, the authors study some fundamental properties on a Min's zeta-function and explore its connection with Hermite elliptic operator.

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On the zeta-function of systems of nonlinear equations

A. M. Kytmanov, S. G. Myslivets

Siberian Mathematical Journal > 2007 > 48 > 5 > 863-870

We introduce the concept of zeta-function for a system of meromorphic functions f = (f ₁,..., f _n) in ℂⁿ. Using residue theory, we give an integral representation for the zeta-function which enables us to construct an analytic continuation of the zeta-function.

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Spectral sets of certain unbounded functions

Yuichi Kamiya

Acta Mathematica Hungarica > 2006 > 110 > 1-2 > 51-65

SummaryA. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of those by trigonometric polynomials. We consider spectral sets of unbounded functions in a certain class which contains the square of the Riemann zeta-function as a typical example.

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Zero Point Energy of a Massless Scalar Field in the Cosmic String Space-Time

Nail R. Khusnutdinov, Artem R. Khabibullin

General Relativity and Gravitation > 2004 > 36 > 7 > 1613-1618

We consider the ground state energy of a massive scalar field in the space-time of a thick cosmic string in the 2+1 dimensional case for arbitrary angle deficit by using the zeta-function approach. Final numerical calculations were made in the massless case, only. We show that the zero point energy is negative, and for small angle deficit it is proportional to the fourth degree of the deficit.

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The Universality of Zeta-Functions

A. Laurinčikas

Acta Applicandae Mathematicae > 2003 > 78 > 1-3 > 251-271

The first part of the paper contains a survey on the universality of zeta-functions. Zeta-functions with Euler's product as well as zeta-functions without Euler's product are discussed. Also, the joint universality theorems are considered. In the second part of the paper the universality of zeta-functions of finite Abelian groups of rank ≤3 is proved.

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Integrals Expressible as Linear Forms in Generalized Polylogarithms

S. A. Zlobin

Mathematical Notes > 2002 > 71 > 5-6 > 711-716

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On the zeta-function of a polynomial at infinity

Sabir M. Gusein-Zade, Ignacio Luengo, Alejandro Melle-Hernández

Bulletin des sciences mathematiques > 2000 > 124 > 3 > 213-224

A polynomial function defines a locally trivial fibre bundle over the complement to a finite set in the target C. Objects connected with this fibration (say, monodromy operators and, in particular, the monodromy operator of the polynomial at infinity) are in some sense global. The idea of the paper is to localize computations of the zeta-functions of monodromy transformations for a polynomial, i.e...

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Regularization schemes and the multiplicative anomaly

T.S. Evans

Physics Letters B > 1999 > 457 > 1-3 > 127-132

Elizalde, Vanzo, and Zerbini have shown that the effective action of two free Euclidean scalar fields in flat space contains a `multiplicative anomaly' when ζ-function regularization is used. This is related to the Wodzicki residue. I show that there is no anomaly when using a wide range of other regularization schemes and that the anomaly can be removed by an unusual choice of renormalization scales...