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13 pages, 1294 KiB

Open AccessArticle

From Complex to Quaternions: Proof of the Riemann Hypothesis and Applications to Bose–Einstein Condensates

by Jau Tang

Symmetry 2025, 17(7), 1134; https://doi.org/10.3390/sym17071134 - 15 Jul 2025

Viewed by 596

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the [...] Read more.

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials: 3rd Edition)

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16 pages, 295 KiB

Open AccessArticle

New Inversion Formulae for the Widder–Lambert and Stieltjes–Poisson Transforms

by Emilio R. Negrín and Jeetendrasingh Maan

Axioms 2025, 14(4), 291; https://doi.org/10.3390/axioms14040291 - 13 Apr 2025

Viewed by 297

Abstract

This paper establishes explicit inversion formulae for the Widder–Lambert transform and the Stieltjes–Poisson transform, extending their applicability to function spaces and compactly supported distributions. Utilizing a generalized Lambert transform, the study provides a unified framework for inversion, enhancing the theoretical understanding of these [...] Read more.

This paper establishes explicit inversion formulae for the Widder–Lambert transform and the Stieltjes–Poisson transform, extending their applicability to function spaces and compactly supported distributions. Utilizing a generalized Lambert transform, the study provides a unified framework for inversion, enhancing the theoretical understanding of these integral transforms. Additionally, a specific function class satisfying Salem’s equivalence to the Riemann hypothesis is identified, further broadening the analytical scope of the results within generalized function spaces. Full article

13 pages, 269 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On Equivalents of the Riemann Hypothesis Connected to the Approximation Properties of the Zeta Function

by Antanas Laurinčikas

Axioms 2025, 14(3), 169; https://doi.org/10.3390/axioms14030169 - 26 Feb 2025

Viewed by 794

Abstract

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

,

m \in N

) lie on the critical line

σ = 1 / 2

. [...] Read more.

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

,

m \in N

) lie on the critical line

σ = 1 / 2

. In this paper, combining the universality property of

ζ (s)

with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts

ζ (s + i t_{τ})

approximating the function

ζ (s)

. Here,

t_{τ}

denotes the Gram function, which is a continuous extension of the Gram points. Full article

12 pages, 261 KiB

Open AccessArticle

Mellin and Widder–Lambert Transforms with Applications in the Salem Equivalence to the Riemann Hypothesis

by Emilio R. Negrín, Jeetendrasingh Maan and Benito J. González

Axioms 2025, 14(2), 129; https://doi.org/10.3390/axioms14020129 - 10 Feb 2025

Cited by 1 | Viewed by 642

Abstract

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we [...] Read more.

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms. Full article

(This article belongs to the Special Issue Elliptic Curves, Modular Forms, L-Functions and Applications)

21 pages, 2287 KiB

Open AccessFeature PaperArticle

Euler–Riemann–Dirichlet Lattices: Applications of η(s) Function in Physics

by Hector Eduardo Roman

Mathematics 2025, 13(4), 570; https://doi.org/10.3390/math13040570 - 9 Feb 2025

Viewed by 808

Abstract

We discuss applications of the Dirichlet

η (s)

function in physics. To this end, we provide an introductory description of one-dimensional (1D) ionic crystals, which are well-known in the condensed matter physics literature, to illustrate the central issue of the paper: [...] Read more.

We discuss applications of the Dirichlet

η (s)

function in physics. To this end, we provide an introductory description of one-dimensional (1D) ionic crystals, which are well-known in the condensed matter physics literature, to illustrate the central issue of the paper: A generalization of the Coulomb interaction between alternating charges in such crystalline structures. The physical meaning of the proposed form, characterized by complex (in the mathematical sense) ion–ion interactions, is argued to have emerged in many-body systems, which may include effects from vacuum energy fluctuations. We first consider modifications to the bare Coulomb interaction by adding an imaginary component to the exponent of the Coulomb law of the form

s = 1 + i b

, where b is a real number. We then extend the results to slower-decaying interactions, where the exponent becomes

s = a + i b

, presenting numerical results for values

1 / 2 \leq a \leq 2

, which include the critical strip relevant to the Riemann hypothesis scenario. Full article

(This article belongs to the Section E4: Mathematical Physics)

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11 pages, 307 KiB

Open AccessReview

A Brief Survey on the Riemann Hypothesis and Some Attempts to Prove It

by Renato Spigler

Symmetry 2025, 17(2), 225; https://doi.org/10.3390/sym17020225 - 4 Feb 2025

Cited by 1 | Viewed by 3221

Abstract

This paper presents a brief survey on the Riemann Hypothesis, a central conjecture in number theory with profound implications, and describes various recent attempts aimed at proving it. Full article

(This article belongs to the Section Mathematics)

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12 pages, 277 KiB

Open AccessFeature PaperArticle

New Analytical Formulas for the Rank of Farey Fractions and Estimates of the Local Discrepancy

by Rogelio Tomás García

Mathematics 2025, 13(1), 140; https://doi.org/10.3390/math13010140 - 2 Jan 2025

Viewed by 1069

Abstract

New analytical formulas are derived for the rank and the local discrepancy of Farey fractions. The new rank formula is applicable to all Farey fractions and involves sums of a lower order compared to the searched one. This serves to establish a new [...] Read more.

New analytical formulas are derived for the rank and the local discrepancy of Farey fractions. The new rank formula is applicable to all Farey fractions and involves sums of a lower order compared to the searched one. This serves to establish a new unconditional estimate for the local discrepancy of Farey fractions that decrease with the order of the Farey sequence. This estimate improves the currently known estimates. A new recursive expression for the local discrepancy of Farey fractions is also given. A second new unconditional estimate of the local discrepancy of any Farey fraction is derived from a sum of the Mertens function, again, improving the currently known estimates. Full article

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8 pages, 728 KiB

Open AccessArticle

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

by Yochay Jerby

Axioms 2024, 13(9), 577; https://doi.org/10.3390/axioms13090577 - 25 Aug 2024

Viewed by 1828

Abstract

The Z-function is the real function given by

Z (t) = e^{i θ (t)} ζ (\frac{1}{2} + i t)

, where

ζ (s)

is the Riemann zeta function, and

θ (t)

is [...] Read more.

The Z-function is the real function given by

Z (t) = e^{i θ (t)} ζ (\frac{1}{2} + i t)

, 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

Z_{N} (t) : = \sum_{k = 1}^{N} \frac{cos (θ (t) - ln (k) t)}{\sqrt{k}}

for any

N \in N

. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE)

Z (t) \approx 2 Z_{\tilde{N} (t)} (t)

for

\tilde{N} (t) = [\sqrt{\frac{t}{2 π}}]

. 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) \approx Z_{N (t)} (t)

for

N (t) = [\frac{t}{2}]

, 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. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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15 pages, 570 KiB

Open AccessArticle

Inverse Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytic Functions

by Sergey K. Sekatskii

Symmetry 2024, 16(9), 1100; https://doi.org/10.3390/sym16091100 - 23 Aug 2024

Cited by 1 | Viewed by 1103

Abstract

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of [...] Read more.

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n,

\frac{d^{n} \ln (f (z))}{d z^{n}} = \frac{d^{n} \ln (g (z))}{d z^{n}}

. The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic

θ_{1}

functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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7 pages, 245 KiB

Open AccessArticle

Remarks on the Connection of the Riemann Hypothesis to Self-Approximation

by Antanas Laurinčikas

Computation 2024, 12(8), 164; https://doi.org/10.3390/computation12080164 - 14 Aug 2024

Cited by 1 | Viewed by 1297

Abstract

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function

ζ (s)

by shifts

ζ (s + i τ)

. In this paper, it is determined [...] Read more.

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function

ζ (s)

by shifts

ζ (s + i τ)

. In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating

ζ (s)

with all but at most countably many accuracies

ε > 0

. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed. Full article

20 pages, 325 KiB

Open AccessArticle

Development of a New Zeta Formula and Its Role in Riemann Hypothesis and Quantum Physics

by Saadeldin Abdelaziz, Ahmed Shaker and Mostafa M. Salah

Mathematics 2023, 11(13), 3025; https://doi.org/10.3390/math11133025 - 7 Jul 2023

Cited by 1 | Viewed by 3559

Abstract

In this study, we investigated a new zeta formula in which the zeta function can be expressed as the sum of an infinite series of delta and cosine functions. Our findings demonstrate that this formula possesses duality characteristics and we established a direct [...] Read more.

In this study, we investigated a new zeta formula in which the zeta function can be expressed as the sum of an infinite series of delta and cosine functions. Our findings demonstrate that this formula possesses duality characteristics and we established a direct connection between the Riemann hypothesis and this new formula. Additionally, we explored the behavior of energy or particles in quantum physics within the proposed mathematical model framework based on the new formula. Our model provides a valuable understanding of several important physics inquiries, including the collapse of the wave function during measurement and quantum entanglement, as well as the double slits experiment. Full article

(This article belongs to the Special Issue New Trends in Special Functions and Applications)

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19 pages, 648 KiB

Open AccessArticle

On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions

by Sergey Sekatskii

Axioms 2023, 12(6), 595; https://doi.org/10.3390/axioms12060595 - 15 Jun 2023

Cited by 2 | Viewed by 1384

Abstract

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties [...] Read more.

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function,

℘_{z} (z,_{} τ)

, leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points

m + n τ

and “demi-lattice” points

m + 1 / 2 + n τ

,

m + (n + 1 / 2) τ

,

m + 1 / 2 + (n + 1 / 2) τ

. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative

θ_{1}' (z | τ)

of the elliptic theta-function and the Weierstrass

ζ

-function, and that far from the origin of coordinates such zeroes of the

ζ

-function tend to the positions of the simple poles of this function. Full article

(This article belongs to the Section Mathematical Analysis)

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7 pages, 251 KiB

Open AccessArticle

On the Order of Growth of Lerch Zeta Functions

by Jörn Steuding and Janyarak Tongsomporn

Mathematics 2023, 11(3), 723; https://doi.org/10.3390/math11030723 - 1 Feb 2023

Viewed by 1577

Abstract

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → [...] Read more.

We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t^13/84+ϵ as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t^ϵ (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros. Full article

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

16 pages, 333 KiB

Open AccessArticle

On the Use of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions for the Calculation of Infinite Sums and the Analysis of Zeroes of Analytical Functions

by Sergey Sekatskii

Axioms 2023, 12(1), 68; https://doi.org/10.3390/axioms12010068 - 7 Jan 2023

Cited by 2 | Viewed by 1821

Abstract

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and [...] Read more.

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. On many occasions, this enables to facilitate the obtaining of known results thus having important methodological meaning. Additionally, some new results, to the best of our knowledge, are also obtained in this way. For example, we established new properties of the sum of inverse zeroes of a digamma function, new formulae for the sums

\sum_{}^{} \frac{k_{i}}{ρ_{i}^{2}}

for zeroes ρ_i of incomplete gamma and Riemann zeta functions having the order k_i (These results can be straightforwardly generalized for the sums

\sum_{}^{} \frac{k_{i}}{ρ_{i}^{n}}

with integer n > 2, and so on.) Full article

(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)

23 pages, 7043 KiB

Open AccessArticle

Flexible Control Strategy for Upper-Limb Rehabilitation Exoskeleton Based on Virtual Spring Damper Hypothesis

by Dezhi Kong, Wendong Wang, Yikai Shi and Lingyun Kong

Actuators 2022, 11(5), 138; https://doi.org/10.3390/act11050138 - 19 May 2022

Cited by 6 | Viewed by 3073

Abstract

The focus of this work is to design a control strategy with the dynamic characteristics of spring damping to realize the virtual flexibility and softness of a rigid-joint exoskeleton without installing real, physical elastic devices. The basic idea of a “virtual softening control [...] Read more.

The focus of this work is to design a control strategy with the dynamic characteristics of spring damping to realize the virtual flexibility and softness of a rigid-joint exoskeleton without installing real, physical elastic devices. The basic idea of a “virtual softening control strategy” for a single rigid joint is that a virtual spring damper (VSD) is installed between the motor and the output shaft. By designing the control signal of the motor, the torque output of the joint actuator is softened so that the output has the characteristics of elasticity and variable stiffness. The transfer velocity profile of human limbs reaching from one posture to another always presents as bell-shaped. According to this characteristic, we constructed a trajectory planning method for a point-to-point position-tracking controller based on a normal distribution function, and it was successfully applied to the control of 5-DoF upper-limb rehabilitation exoskeleton. A multi-joint cooperative flexible controller based on the virtual spring damper hypothesis (VSDH) was successfully applied to solve the constrained control problem of the exoskeletons and the self-motion problem caused by redundant degrees of freedom (DoFs). The stability of the closed-loop controlled system is theoretically proven by use of the scalar energy function gradient method and the Riemann metric convergence analysis method. Full article

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