Riemann Zeta Research Articles

The average number of Goldbach representations and Zero-free regions of the Riemann zeta function

In this paper, we prove an unconditional form of Fujii’s formula for the average number of Goldbach representations and show that the error in this formula is determined by a general zero-free region of the Riemann zeta function, and vice versa. In particular, we describe the error in the unconditional formula in terms of the remainder in the Prime Number Theorem which connects the error to zero-free regions of the Riemann zeta function.

A Billiard in an Open Circle and the Riemann Zeta Function

We consider a dynamical billiard in a circle with one or two holes in the boundary, or q symmetrically placed holes. It is shown that the long-time survival probability, either for a circle billiard with discrete or with continuous time, can be written as a sum over never-escaping periodic orbits. Moreover, it is demonstrated that in both cases the Mellin transform of the survival probability with respect to the hole size has poles at locations determined by zeros of the Riemann zeta function and, in some cases, Dirichlet L functions.

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.

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.

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.

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.

A motivic pairing and the Mellin transform in function fields

We define two pairings relating the A-motive with the dual A-motive of an abelian Anderson A-module. We show that specializations of these pairings give the exponential and logarithm functions of this Anderson A-module, and we use these specializations to give precise formulas for the coefficients of the exponential and logarithm functions. We then use these pairings to express the exponential and logarithm functions as evaluations of certain infinite products. As an application of these ideas, we prove an analogue of the Mellin transform formula for the Riemann zeta function in the case of Carlitz zeta values. We also give an example showing how our results apply to Carlitz multiple zeta values.

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

Apocalyptic Proof of the Riemann Hypothesis with the Generating Function for Prime Numbers

The Riemann zeta function ζ(s) plays a crucial role in number theory and its applications such as cryptography. By utilizing the implications of the hypothesis and the distribution of prime numbers, algorithms can be created using primes to send and receive data with reliable security. The Riemann Hypothesis (RH) posits that zeros of ζ(s) other than the trivial ones are located on the line defined by the equation Re(s) =1/2. This paper introduces a novel and straightforward proof of the Riemann Hypothesis. The proof employs a standard method, utilizing the eta function in place of the zeta function, under the assumption that the real part is greater than zero. The equation for the real and imaginary parts of the Riemann zeta function (eta function) is completely separated. Initially, let ζ(s) = Reζ(s) + Imζ(s) with s = a + ib. The value of the real part is determined by solving the equation, and the process is repeated for ζ(1 - s) identifying the potential roots shared by the two functions, a common value is obtained, leading to a =1/2, which represents the real part of the main root of the function ζ(s). Using a standard method and with the help of two functions ζ(s) and ζ(1-s), the real part of the root of the zeta function is obtained. To create a generator function for prime numbers in terms of b, one can solve the root of the zeta function where it equals one (i.e., ζ(s) =1 ) and obtain a relationship between b’ and prime numbers. Giving the value of zeta equal to one and s' = a' + ib’, similar to zeta equal to zero, the roots are again placed on the 1/2 line. Then, by using the zeta function defined by multiplying prime numbers, we arrive at a new meaningful relation between b’ and its corresponding prime number.

Sign Changes of the Error Term in the Piltz Divisor Problem

Abstract We study the function $\Delta _{k}(x):=\sum _{n\leq x} d_{k}(n) - \mbox{Res}_{s=1} ( \zeta ^{k}(s) x^{s}/s )$, where $k\geq 3$ is an integer, $d_{k}(n)$ is the $k$-fold divisor function, and $\zeta (s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindelöf hypothesis (LH) is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon }$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon }$, such that $|\Delta _{k}(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. In particular, $\Delta _{k}(x)$ does not change sign in any of these subintervals. If the Riemann hypothesis (RH) is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^{2}-2}$. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of $\Delta _{k}(x+h)-\Delta _{k}(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $\Delta _{k}(x)$, which we obtain by combining a method of Tsang with a technique of Lester.

Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions

Abstract Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.