Zeta Function Research Articles

A generalized Selberg zeta function for flat space cosmologies

Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}/ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results.

The study of generalized Hurwitz–Lerch zeta function and fractional kinetic equations

The study of generalized Hurwitz–Lerch zeta function and fractional kinetic equations

HIGHER POWER MOMENTS OF COEFFICIENTS ATTACHED TO THE DEDEKIND ZETA FUNCTION

Let K3 be a non-normal cubic extension over Q. In this paper, we investigate the higher moments of coefficients aK3 (n) of the Dedekind zeta function of the following type X n≤x al K3 (n), where l ≥ 4 is any fixed positive integer.

Exploring Explicit Definite Integral Formulae with Trigonometric and Hyperbolic Functions

Making use of integration by parts and variable replacement methods, we derive some interesting explicit definite integral formulae involving trigonometric or hyperbolic functions, whose results are expressed in terms of Catalan’s constant, Dirichlet’s beta function, and Riemann’s zeta function, as well as π in the denominator.

Violation of the third law of thermodynamics by black holes, Riemann zeta function and Bose gas in negative dimensions

Violation of the third law of thermodynamics by black holes, Riemann zeta function and Bose gas in negative dimensions

The modularity of Siegel’s zeta functions

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel’s original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that “half” of Siegel’s zeta functions correspond to holomorphic modular forms.

Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives

LetζE(s,q)=∑n=0∞(−1)n(n+q)s be the alternating Hurwitz (or Hurwitz-type Euler) zeta function.In this paper, we obtain the following asymptotic expansion of ζE(s,q)ζE(s,q)∼12q−s+14sq−s−1−12q−s∑k=1∞E2k+1(0)(2k+1)!(s)2k+1q2k+1, as |q|→∞, where E2k+1(0) are the special values of odd-order Euler polynomials at 0, and we also consider representations and bounds for the remainder of the above asymptotic expansion. In addition, we derive the asymptotic expansions for the higher order derivatives of ζE(s,q) with respect to its first argumentζE(m)(s,q)≡∂m∂smζE(s,q), as |q|→∞. Finally, we also prove a new exact series representation of ζE(s,q).

Statistical Study of Euler’s Zeta Function by using Riemann Hypothesis

The aim of this paper to study the statistical measures for the zeta function and the relative comparison of the function. Here we give an examples of some of the suggested models and compare these models. The development of these models were done in a collection of machine learning algorithms for data mining tasks. As a measure of success of any model, we study the statistical measures CC, MAE, RMSE, RAE and RRSE. We would like to mention that there is no specific rational behind the choice of the statistical measures studied; except that they give good point-wise and overall statistics on the validity and success of the models. The statistical measures Root Mean Square Error and Mean Absolute Error are concerned with the average difference errors.

Representation zeta function of a family of maximal class groups: Non-exceptional primes

Abstract We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups M n M_{n} with maximal nilpotency class. For representation dimensions coprime to all primes p < n p<n , we construct all irreducible representations of M n M_{n} by defining a standard form for the matrices of these representations.

On the Sums over Inverse Powers of Zeros of the Hurwitz Zeta Function and Some Related Properties of These Zeros

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.

The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function

Abstract On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $1/2+i\gamma$ of the Riemann zeta function, we show that the sequence \begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a<b$ , and $m_\gamma$ denotes the multiplicity of the zero $1/2+i{\gamma }$ . The same result holds when the ${\gamma }$ ’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $\gamma (\!\log T)/2\pi$ with ${\gamma }\in \Gamma_{[a, b]}$ and $0<{\gamma }\leq T$ .

Power-series solution of the L-fractional logistic equation

We consider the L-fractional derivative, which has been proposed in the literature to study fractional differentials in geometry and processes in mechanics. Our context is population growth and epidemiology, for which the use of L-derivatives is motivated by transitions. Using power series, we solve the logistic differential equation model under this fractional derivative. Several conclusions on the method, the derivative, and the singularity of the associated kernel are reached. Fractional Euler numbers, related to the logistic map and the famous Riemann zeta function, are also introduced.

A quantization of interacting particle systems

Interacting particle systems studied in this paper are probabilistic cellular automata with nearest-neighbor interaction including the Domany-Kinzel model. A special case of the Domany-Kinzel model is directed percolation. We regard the interacting particle system as a Markov chain on a graph. Then we present a new quantization of the interacting particle system. After that, we introduce a zeta function of the quantized model and give its determinant expression. Moreover, we calculate the absolute zeta function of the quantized model for the Domany-Kinzel model.

Universality theorems of Selberg zeta functions for arithmetic groups

Abstract We prove a universality theorem for the Selberg zeta function of subgroups of $\mathrm{SL}_2(\mathbb{Z})$ or co-compact arithmetic groups derived from quaternion algebras, in the strip $\{5/6 \lt \mathrm{Re}{s} \lt 1\}$, improving the range compared with a previous work by Drungilas–Garunkštis–Kačenas. We also obtain the same range for a joint universality theorem for congruence subgroups, which improves a result by Mishou.

Asymptotic expansions for partitions generated by infinite products

AbstractRecently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $$\Lambda \subset {\mathbb {N}}$$ Λ ⊂ N ($$\gcd (\Lambda )=1$$ gcd ( Λ ) = 1 ) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $$\Lambda $$ Λ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree n of $${\mathfrak {so}{(5)}}$$ so ( 5 ) . We also study the Witten zeta function $$\zeta _{{\mathfrak {so}{(5)}}}$$ ζ so ( 5 ) , which is of independent interest.

On Universality of Some Beurling Zeta-Functions

Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζP(s+iτ), τ∈R. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set {logp:p∈P}, and the existence of a bounded mean square for ζP(s). Under the above hypotheses, we obtain the universality of the function ζP(s). This means that the set of shifts ζP(s+iτ) approximating a given analytic function defined on a certain strip σ^<σ<1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied.

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a D-dimensional flat space under the influence of a shell potential. The static nature of the potential, and the spherical symmetry, allows us to focus on the spatial part of the field which satisfies a one-dimensional Schrodinger equation endowed with a point potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the one-dimensional Schrodinger equation that describes the radial part of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

Pseudo-differential operators on the circle, Bernoulli polynomials

We show how the classical polylogarithm function Lis(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Li}_s(z)$$\\end{document} and its relatives, the Hurwitz zeta function and the Lerch function are all of a spectral nature, and can explain many properties of the complex powers of the Laplacian on the circle and of the distribution (x+i0)s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x+i 0)^s$$\\end{document}. We also make a relation with a result of Keiper [Fractional Calculus and its relationship to Riemann’s zeta function, Master of Science, Ohio State University, Mathematics (1975)].

Identities among some combinatorial objects involving special values of multiple zeta functions

In the article, we establish some identities involving special values of multiple zeta functions among the counting functions of number of representations of an integer by a linear combination of figurate numbers such as triangular numbers, square numbers, pentagonal numbers, etc. More precisely, we provide our result for $\delta_k(n)$, $r_{k}(n)$ and $\mathcal{N}_{k}^{a}(n)$ (for a fixed $a \ge 3$), the number of representations of $n$ as a sum of $k$-triangular numbers, as a sum of $k$-square numbers and as a sum of $k$-higher figurate numbers (for a fixed $a \ge 3$), respectively. Moreover, these identities also occur when one of $\delta_k(n)$, $r_{k}(n)$ and $\mathcal{N}_{k}^{a}(n)$ is replaced by the $k$-colored partition functions.

Zeta functions in higher Teichmüller theory

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in PSL(d,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PSL}(d, \\mathbb R)$$\\end{document}, for d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}. These are natural generalizations of the well known classical Selberg zeta function and L-function for Fuchsian groups, corresponding to the case d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}. We show that these complex functions have meromorphic extensions to the entire complex plane C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb C$$\\end{document}.