Zeta Function Research Articles

Infinite order linear difference equation satisfied by a refinement of Goss zeta function

Infinite order linear difference equation satisfied by a refinement of Goss zeta function

A further generalization of sums of higher derivatives of the Riemann zeta function

We obtain a full asymptotic for the sum of [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th derivative of the Riemann zeta function, [Formula: see text] is a positive real number, and [Formula: see text] denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height [Formula: see text], as [Formula: see text]. We also specify what the asymptotic formula becomes when [Formula: see text] is a positive integer, highlighting the differences in the asymptotic expansions as [Formula: see text] changes its arithmetic nature.

Collatz Conjecture

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.This analysis has allowed to understand that the Collatz sequence initially tends towards a balanced symmetric arrangement of zeros and ones, and then it collapses, realizing the scenario of the Collatz conjecture.

Twisted mixed moments of the Riemann zeta function

AbstractWe analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape for a suitable constant and a polynomial . Such examinations are performed both unconditionally and under the assumption of a weaker version of the ‐conjecture.

Riemann Zeta Function and Its Applications

Abstract. This paper focuses on the introduction and proof of the fundamental properties of (z), i.e. Riemann zeta function and explores its applications in algebra. We begin with a systematic derivation and proof of the basic characteristics of the zeta function. Following this, we examine its application in algebra, including the use of the Dirichlet L-function to prove Dirichlets theorem. Furthermore, we show the classical result for the subgroup growth rate of J-groups and the enumeration of n-dimensional irreducible representations of Heisenberg groups.

A reciprocity relation for the twisted second moment of the Riemann Zeta function

We prove a reciprocity relation for the twisted second moment of the Riemann Zeta function. This provides an analogue to a formula of Conrey for Dirichlet L L -functions.

On the computation of lattice sums without translational invariance

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics and have immediate applications in condensed matter and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools like Ewald summation ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method’s runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient and precise simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with 3 × 10 23 3\times 10^{23} particles. Our method’s accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.

Modeling uncertainty with the truncated zeta distribution in mixture models for ordinal responses

In recent three decades, there has been a rapid increasing interest in the mixture models for ordinal responses, and the classical CUB model as a fundamental one has been extended to different preference and uncertainty models. In this article, based on a response style supported by Zipf’s law, we propose a novel mixture model for ordinal responses via replacing the uncertainty component of the CUB model with a truncated Zeta distribution. Parameters estimation with EM algorithm, inferential issues with respect to the approximation of a truncated Riemann Zeta function and estimators’ variance-covariance information matrix are investigated. The advantages of the proposed model over the CUB model have been illustrated with simulations of two sets of Monte Carlo experiments and practical applications of a health survey and a bicycle use. The intention of the article is to distinguish the respondents’ true preference from the response style of “the higher, the less”, so as to understand more reasonably the formation causes of ordinal responses.

Higher Jacobian ideals, contact equivalence and motivic zeta functions

We show basic properties of higher Jacobian matrices and higher Jacobian ideals for functions and apply it to obtain two main results concerning singularities of functions. Firstly, we prove that a higher Nash blowup algebra is invariant under contact equivalences, which was recently conjectured by Hussain, Ma, Yau and Zuo. Secondly, we obtain an analogue of a result on motivic nearby cycles by Bussi, Joyce and Meinhardt.

Linear Arrangement of Euler Sums with Multiple Argument

We investigate the linear arrangement of Euler harmonic sums that may be expressed in closed form in terms of special functions such as the classical Riemann zeta function and the Dirichlet eta function. Particular emphasis is given to Euler harmonic sums with even weight. New examples highlighting the theorems will be presented.

Quadratic enrichment of the logarithmic derivative of the zeta function

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

Absolute Zeta functions and periodicity of quantum walks on cycles

The quantum walk is a quantum counterpart of the classical random walk. On the other hand, absolute zeta functions can be considered as zeta functions over $\mathbf{F}_1$. This study presents a connection between quantum walks and absolute zeta functions. In this paper, we focus on Hadamard walks and $3$-state Grover walks on cycle graphs. The Hadamard walks and the Grover walks are typical models of the quantum walks. We consider the periods and zeta functions of such quantum walks. Moreover, we derive the explicit forms of the absolute zeta functions of corresponding zeta functions. Also, it is shown that our zeta functions of quantum walks are absolute automorphic forms.

A note on the existence of the Reidemeister zeta function on groups

Given an endomorphism φ:G→G on a group G, one can define the Reidemeister number R(φ)∈N∪{∞} as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function Rφ(z), by using the Reidemeister numbers R(φn) of iterates φn in order to define a power series, has been studied a lot in the literature, especially the question whether it is a rational function or not. For example, it has been shown that the answer is positive for finitely generated torsion-free virtually nilpotent groups, but negative in general for abelian groups that are not finitely generated.However, in order to define the Reidemeister zeta function of an endomorphism φ, it is necessary that the Reidemeister numbers R(φn) of all iterates φn are finite. This puts restrictions, not only on the endomorphism φ, but also on the possible groups G if φ is assumed to be injective. In this note, we want to initiate the study of groups having a well-defined Reidemeister zeta function for a monomorphism φ, because of its importance for describing the behavior of Reidemeister zeta functions. As a motivational example, we show that the Reidemeister zeta function is indeed rational on torsion-free virtually polycyclic groups. Finally, we give some partial results about the existence in the special case of automorphisms on finitely generated torsion-free nilpotent groups, showing that it is a restrictive condition.

A discrete mean value of the Riemann zeta function

AbstractIn this work, we estimate the sum over the nontrivial zeros of the Riemann zeta function where is a complex number with and and are Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where is replaced by for all . The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann hypothesis, we obtain the lower bound which was previously known under the generalized Riemann hypothesis, only in the case .

Inequalities for Basic Special Functions Using Hölder Inequality

Let p,q≥1 be two real numbers such that 1p+1q=1, and let a,b∈R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form |f(ap+bq)|≤|f(a)|1p|f(b)|1q, and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities.

Emergent family of Tsallis entropies from the q-deformed combinatorics

We revisit the derivation of a formula for the q-generalised multinomial coefficient rooted in the q-deformed algebra, a foundational framework in the study of nonextensive statistics. Previous approximate expressions in the literature diverge as q approaches 2 (or 0, depending on convention). In contrast, our derived formula provides an exact, smooth function for all real values of q, expressed as an infinite series expansion involving Tsallis entropies with sequential entropic indices, coupled with Bernoulli numbers. This formulation is achieved through the analytic continuation of the Riemann zeta function, stemming from the q-deformed factorials. Our formula thus offers a distinctive characterisation of Tsallis entropy within the q-deformed combinatorics. Throughout this exploration, we also highlight a symmetry within the q-deformed theory that links different values of the entropic parameter. Furthermore, we discuss extending our results to encompass more general, affinity-sensitive cases, building on the previously established framework of affinity-based extended entropy.

Series over Bessel Functions as Series in Terms of Riemann’s Zeta Function

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions.

A Family of Integrals Related to Values of the Riemann Zeta Function

A Family of Integrals Related to Values of the Riemann Zeta Function

Information, Entropy, and the Zeta Function

Information, Entropy, and the Zeta Function

A census of cubic fourfolds over 𝔽₂

We compute a complete set of isomorphism classes of cubic fourfolds over F 2 \mathbb {F}_2 . Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all zeta functions of the smooth cubic fourfolds over F 2 \mathbb {F}_2 ; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over F 2 \mathbb {F}_2 , which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of K 3 K3 surfaces. We present a refinement to the standard method of orbit enumeration that leverages filtrations and gives a significant speedup. In the case of cubic fourfolds, the relevant filtration is determined by Waring representation and the method brings the problem into the computationally tractable range.