Zeta Function Research Articles

Recurrence formulae for spectral determinants

We develop a unified method to study spectral determinants for several different manifolds, including spheres and hemispheres, and projective spaces. This is a direct consequence of an approach based on deriving recursion relations for the corresponding zeta functions, which we are then able to solve explicitly. Apart from new applications such as hemispheres, we also believe that the resulting formulae in the cases for which expressions for the determinant were already known are simpler and easier to compute in general, when compared to those resulting from other approaches.

Asymptotic Expansions for the Radii of Starlikeness of Normalized q-Bessel Functions

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials.

On Closed Forms of Some Trigonometric Series

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative.

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.

Half-Isolated Zeros and Zero-Density Estimates

Abstract We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few “half-isolated” zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.

Discrete Ω-results for the Riemann zeta function

Abstract We study lower bounds for the Riemann zeta function ζ ⁢ ( s ) {\zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ ⁢ ( s ) {\zeta(s)} .

Some identities on degenerate harmonic and degenerate higher-order harmonic numbers

The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this paper is to study the degenerate harmonic and degenerate higher-order harmonic numbers, which are respectively degenerate versions of the harmonic and higher-order harmonic numbers, in connection with the degenerate zeta and degenerate Hurwitz zeta function. Here the degenerate zeta and degenerate Hurwitz zeta function are respectively degenerate versions of the Riemann zeta and Hurwitz zeta function. We show that several infinite sums involving the degenerate higher-order harmonic numbers can be expressed in terms of the degenerate zeta function. Furthermore, we demonstrate that an infinite sum involving finite sums of products of the degenerate harmonic numbers can be represented by using the degenerate Hurwitz zeta function.

A Linear Relation for Values of the Zeta Function at Even Positive Integers

A Linear Relation for Values of the Zeta Function at Even Positive Integers

Light-matter Interaction and Zeta Functions

Light-matter Interaction and Zeta Functions

Irrational Outputs from Rational Inputs in Theta-3 Functions Using Dirichlet’s Theorem

Theta Functions are not an easy subject of mathematics because new researches about this wide topic are still being published regularly. The findings about Theta Functions lead directly to many applications in different fields of applied mathematics and help also to develop old interesting mathematical topics such as Poisson summation and Riemann Zeta Function. This work proves by using a theorem of Dirichlet and a previously demonstrated useful approximation that the rational inputs of Theta-3 Functions give only irrationals as outputs of these functions. This paper applies only simple notions of sequences and represents an opportunity for the students of mathematics to easily understand the steps of the demonstrations.

Deriving measurement collapse using zeta function regularisation and speculative measurement theory

This paper shows how an application of zeta function regularisation to a physical model of quantum measurement yields a solution to the problem of wavefunction collapse. Realistic measurement dynamics based on a particle becoming non-isolated are introduced and, based on this, an outcome function is derived using the method of maximum entropy. It is shown how regularisation of an information theoretic quantity related to this outcome function leads to apparent collapse of the wavefunction. The physical principles and key assumptions that underlie this theory are discussed. Some possible experimental approaches are described.

Rationality of twist representation zeta functions of compact 𝑝-adic analytic groups

We prove that for any twist rigid compact p p -adic analytic group G G , its twist representation zeta function is a finite sum of terms n i − s f i ( p − s ) n_{i}^{-s}f_{i}(p^{-s}) , where n i n_{i} are natural numbers and f i ( t ) ∈ Q ( t ) f_{i}(t)\in \mathbb {Q}(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G G is moreover a pro- p p group, we prove that its twist representation zeta function is rational in p − s p^{-s} . To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.

Zeta Functions and the (Linear) Logic of Markov Processes

Zeta Functions and the (Linear) Logic of Markov Processes

Generalized L-functions related to the Riemann zeta function

Generalized L-functions related to the Riemann zeta function

A smooth version of Landau’s explicit formula

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number [Formula: see text] is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order [Formula: see text]. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.

On the Approximation of the Hardy Z-Function via High-Order Sections

The Z-function is the real function given by Z(t)=eiθ(t)ζ12+it, where ζ(s) is the Riemann zeta function, and θ(t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z(t) and its zeros. The sections of Z(t) are given by ZN(t):=∑k=1Ncos(θ(t)−ln(k)t)k for any N∈N. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z(t)≈2ZN˜(t)(t) for N˜(t)=t2π. While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z(t)≈ZN(t)(t) for N(t)=t2, which is Spira’s approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira’s conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics.

An Approximation of the Prime Counting Function and a New Representation of the Riemann Zeta Function

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuable. These methods also offer avenues for number-theoretic exploration through analytical manipulation. In this work, we introduce a novel approximation function, ϕ(n), which adds to the existing repertoire of approximation methods and provides a fresh perspective for number-theoretic studies. Deeper analytical investigation of ϕ(n) reveals modified representations of the Chebyshev function, prime number theorem, and Riemann zeta function. Computational studies indicate that the difference between ϕ(n) and the logarithmic integral at magnitudes greater than 10100 is less than 1%.

Subconvexity of twisted Shintani zeta functions

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip.

C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function

C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function