Number Theory Research Articles

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.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

Integer partitions detect the primes

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n ≥ 2 is prime if and only if [Formula: see text]where the [Formula: see text] are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions [Formula: see text] we prove that there are infinitely many such prime detecting equations with constant coefficients, such as [Formula: see text].

RADIAL ASYMPTOTICS OF GENERATING FUNCTIONS OF k-REGULAR SEQUENCES

AbstractWe give a new proof of a theorem of Bell and Coons [‘Transcendence tests for Mahler functions’, Proc. Amer. Math. Soc.145(3) (2017), 1061–1070] on the leading order radial asymptotics of Mahler functions that are the generating functions of regular sequences. Our method allows us to provide a description of the oscillations whose existence was shown by Bell and Coons. This extends very recent results of Poulet and Rivoal [‘Radial behavior of Mahler functions’, Int. J. Number Theory, to appear].

Matemáticas en el mundo árabe: Un recorrido por la edad de oro del saber

The history of Arabic mathematics is crucial to the development of mathematics. During the Islamic Golden Age (8th-14th centuries), Arab mathematicians made important contributions in algebra, geometry, trigonometry, and arithmetic, applying their knowledge in astronomy, navigation, and engineering. A significant advance was the introduction of the Hindu-Arabic numeral system, which facilitated arithmetic operations. Prominent mathematicians such as Al-Khwarizmi, who developed methods for solving equations and wrote about algebra, and Omar Khayyam, known for his work in algebraic calculus, made fundamental contributions. Thabit ibn Qurra also contributed to geometry and number theory. In addition, Arabic mathematics preserved and translated works of ancient civilizations, which allowed the transmission of this knowledge to Europe and its influence on the Renaissance.

Historical Analysis Of The Development Of Number Theory In The Islamic Intellectual Tradition

This article explores the contributions of Muslim scholars to the development of number theory and their impact on global mathematics. Through a qualitative research method with a library research approach, the study examines the works of notable figures such as Al-Khwarizmi, Al-Kindi, Omar Khayyam, and Al-Karaji, highlighting their roles in refining and advancing mathematical concepts. It also discusses the philosophical and theological influences that shaped their approaches to numbers and mathematics. The article further analyzes the transmission of these ideas to the Western world and their subsequent influence on Western mathematical traditions. The findings suggest that Muslim scholars played a pivotal role in the evolution of number theory by integrating knowledge from various civilizations and introducing innovative mathematical frameworks. Despite certain limitations regarding source accessibility and reliance on historical interpretations, the research underscores the enduring impact of these contributions on the development of modern mathematics.

Advance in Applied Cryptography Theory: Survey and New Results. Part 1. Key Cryptography

In the current paper, consisting from two parts, are presented both results already published before (but hard for access) and new once. Actuality of this work is firstly in a fact that recently has been obtained a number of new results in area of applied cryptography that are needed both in a clarification and be put into practice. This is namely the main goal of the current paper. The setting problem in the first paper part concerns to a complexity of symmetric cipher breaking while in the second part of the paper is discussed, so called, keyless cryptography, namely: wiretap channel concept, execution of communication channels which allow to provide information security without of key exchange procedure between legal correspondences. In the part widely used methods of applied mathematics, namely: algebra, number, probability and information theories. Computer simulation also used there. A novelty of the first part of work consists in the following: first of all it is clarified the sense of a key lifetime limitation for different symmetric cipher modes, secondly, it is explained an approach of cipher breaking by the use of quantum computers, finely, the key authentication for the Diffie ‒ Hellman protocol based on the mobile device pairing technology is investigated in detail. In the second part of the current paper has been presented a vulnerability of Dean ‒ Goldsmith cryptosystem under some extension of attacks. The main results of this paper are: estimation of the key lifetime of single key for symmetric cipher in CBC mode, clarifying of Grover’s algorithm breaking of symmetric ciphers by brute force attack, development of a method for authentication of Diffie ‒ Hellman values based on pre-distributed sequences, selection of ciphers which allow to execute with Shamir’s protocol without any key sharing in advance, breaking of Dean ‒ Goldsmith protocol under some conclusions, proof the fact regarding of a possible breakability of the key sharing protocol over noiseless communication channels. Practical application of paper results consists in the fact of stimulation the correct choice of ciphers and their parameters in order to provide their resistance to different attacks and more attention to algorithms of keyless cryptography.

LB-CLAS: Lattice-based conditional privacy-preserving certificateless aggregate signature scheme for VANET

The rapid development of vehicular ad-hoc networks (VANETs) has brought great convenience to intelligent transportation, and the secure transmission of information in VANETs has become a serious problem. In addition, the protection of private information of vehicles is also a key issue. Aiming at the problem of how to guarantee the secure transmission of information in VANETs under the condition of satisfying security and privacy, we propose a lattice-based conditional privacy-preserving certificateless aggregate signature scheme (LB-CLAS) for VANETs. Instead of using Number Theory Research Unit (NTRU) lattices and discrete Gaussian sampling, the proposed LB-CLAS scheme is based on algebraic lattice. In addition, based on the module version of Small Integer Solution (MSIS) and module version of Learning With Error (MLWE) hard problems, we prove that the LB-CLAS scheme is existential unforgeability under adaptively chosen message attacks (EUF-CMA). Our LB-CLAS scheme employs individual signature verification in vehicle-to-vehicle (V2V) mode, while utilizing aggregate signatures and batch verification in vehicle-to-infrastructure (V2I) mode, with slightly differing transmission parameters between the two modes. Based on Dilithium, our LB-CLAS scheme solves the problem of high storage overhead and computational cost of existing schemes. The performance analysis shows that our LB-CLAS scheme is more efficient in terms of computation cost, storage overhead, and power consumption compared to existing schemes. Compared with existing schemes, our LB-CLAS scheme reduces the signature and verification overheads by more than 17.6% and 43.4%, respectively. Our LB-CLAS program also has significant advantages in batch verification. As the number of vehicles increases, our batch certification time cost is reduced by more than 90%. In addition, our LB-CLAS scheme has the smallest signature length, with a signature size that is 1X smaller than the most efficient existing scheme for the same level of security.

Prime number factorization and degree of coherence of speckled light beams.

We discover a connection between a Gauss sum of number theory and the degree of coherence (DOC) of the field in a transverse plane of structured speckled light beams. We theoretically demonstrate and experimentally validate that prime number factorization can be achieved by manipulating the source beam's DOC in Young's double-slit experiment. The determination of whether a number can be factored is based solely on the visibility of the resulting interference patterns. Our findings offer new insights into information encryption and decryption, data compression, etc.

Arithmetic branching law and generic 𝐿-packets

Let G G be a classical group defined over a local field F F of characteristic zero. For any irreducible admissible representation π \pi of G ( F ) G(F) , which is of Casselman-Wallach type if F F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F F . In particular, if π \pi has a generic local L L -parameter, we introduce the spectral first occurrence index f s ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi ) and the arithmetic first occurrence index f a ( π ) {\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) of π \pi and prove in this paper that f s ( π ) = f a ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi )={\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) . Based on the theory of consecutive descents of enhanced L L -parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L L -packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality.

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 Mathematical World Woven by Number Theory, Algebraic Geometry, and Representation Theory

A Mathematical World Woven by Number Theory, Algebraic Geometry, and Representation Theory

How Number Theory Elucidates the Mysteries of Complex Dynamics--Viewed through Non-Archimedean Dynamics

How Number Theory Elucidates the Mysteries of Complex Dynamics--Viewed through Non-Archimedean Dynamics

Foundations of iterated star maps and their use in combinatorics

We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.

Generalized circular impact time guidance

To effectively control impact time and ensure terminal acceleration converges to zero under the look angle constraint, this paper proposes a generalized circular impact time guidance law, providing a deeper insight into circular guidance. The proposed guidance law is formulated based on a generalized circular guidance equation that offers an explicit solution for the time to go. The convergence of both miss-distance and terminal acceleration to zero is proven using Bernoulli number theory. Impact time control is achieved by employing a fixed-time convergent controller that guides the look angle to track desired values. Additionally, the look angle constraint is addressed by incorporating a well-designed function into the acceleration. The proposed guidance law does not require estimated time to go and small-angle assumptions for its implementation. By predicting the mean speed, the proposed guidance law remains valid for missiles with varying speeds. Furthermore, the planar results are extended to three-dimensional scenarios by adopting the concept of the engagement plane. Nonlinear simulations demonstrate the effectiveness, advantages, and robustness of the proposed guidance law in the presence of disturbances and autopilot lag.

A Review on Ancient Cryptographic, Modern Cryptographic and Quantum Cryptographic Techniques

Cryptography has a long and fascinating history, evolving from ancient techniques to modern methods and now exploring the potential of quantum mechanics. This review paper provides a comprehensive overview of cryptographic techniques from past to present. We begin by examining ancient cryptographic techniques, tracing their origins back to 2000 B.C. when the ancient Egyptians used "secret" hieroglyphics. We also discuss evidence from ancient Greece and Rome, such as secret writings and the famous Caesar cipher [1]. These early methods laid the foundation for the field of cryptography. Next, we delve into modern cryptographic techniques that have become increasingly complex and diverse in their applications. We explore how cryptography now makes extensive use of mathematical concepts from fields like information theory, computational complexity, statistics, combinatorics, abstract algebra, number theory, and finite mathematics. We also discuss how the development of digital computers and electronics has revolutionized cryptography, allowing for the encryption of any kind of binary data and the design of much more intricate ciphers [2]. Finally, we examine the emerging field of quantum cryptography, which aims to harness the principles of quantum mechanics to achieve new cryptographic functionalities beyond classical information alone. We survey some of the most remarkable theoretical uses of quantum information for cryptography, as well as the limitations and challenges faced by cryptographers in this new paradigm [3]. By tracing the evolution of cryptography from ancient times to the present day and beyond, this review paper provides valuable insights into the rich history and promising future of this critical field of study.

Local Non Abelian Class Field Theory

The “local class field theory”, which can be defined as the description of the extensions of a given local field K in terms of the algebraic and analytic objects depending ony on the base K is one of the central problems of modern number theory. The theory developed for the abelian extensions, around the fundamental works of Artin and Hasse in the first quarter of the 20th century. It is natural to ask if one could construct this theory including the non-abelian extensions of the base field. There are two approches to this problem. One approach is based on the ideas of Langlands, and the other on Koch. Koch’s method was later generalized by Fesenko and Koch-de Shalit for some non-abelian extensions of the base field. On the other hand, Ikeda and Serbest extended Fesenko’s works to construct a non-abelian local class field theory. But there is a restriction for the base field K. In this study, we extended Ikeda-Serbest’s construction of the local reciprocity map for a certain local field to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties.

Mathematical analysis of isothermal study of reverse roll coating using Micropolar fluid

This article demonstrates a mathematical model and theoretical analysis of the Micropolar fluid in the reverse roll coating process. It is important because micropolar fluids account for the microstructure and microrotation of particles within the fluid. These characteristics are significant for accurately describing the behavior of complex fluids such as polymer solutions, biological fluids, and colloidal suspensions. First, we modeled the flow equations using basic laws of fluid dynamics. The flow equations are made modified using low Reynolds number theory. The simplified equations are solved analytically. The exact expression for velocity and pressure gradient are obtained, while pressure is calculated numerically using Simpson Rule. Graphical depictions are carried out to comprehend the impact of the newly emerged physical constraints. The influence of micropolar and microrotation parameters on the velocity, pressure and pressure gradient are elaborated with the help of different graphs.

X-y duality in topological recursion for exponential variables via quantum dilogarithm

For a given spectral curve, the theory of topological recursion generates two different families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ of multi-differentials, which are for algebraic spectral curves related via the universal x-yx−y duality formula. We propose a formalism to extend the validity of the x-yx−y duality formula of topological recursion from algebraic curves to spectral curves with exponential variables of the form e^x=F(e^y)ex=F(ey) or e^x=F(y)e^{a y}ex=F(y)eay with FF rational and aa some complex number, which was in principle already observed in [Commun. Number Theory Phys. 13, 763 (2019); J. Lond. Math. Soc. 109, e12946 (2024)]. From topological recursion perspective the family \omega_{g,n}^\veeωg,n∨ would be trivial for these curves. However, we propose changing the n=1n=1 sector of \omega_{g,n}^\veeωg,n∨ via a version of the Faddeev’s quantum dilogarithm which will lead to the correct two families \omega_{g,n}ωg,n and \omega_{g,n}^\veeωg,n∨ related by the same x-yx−y duality formula as for algebraic curves. As a consequence, the x-yx−y symplectic transformation formula extends further to important examples governed by topological recursion including, for instance, Gromov-Witten invariants of \mathbb{C}^3ℂ3 (or, equivalently, triple Hodge integrals), orbifold Hurwitz numbers, and stationary Gromov-Witten invariants of \mathbb{P}^1ℙ1. The proposed formalism is related to the issue topological recursion encounters for specific choices of framings for the topological vertex curve.

On the embedding of Galois groups into wreath products

In this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields F ⊆ K ⊆ L with L/F finite and separable, we explicitly construct an embedding of the Galois group Gal ( L c / F ) into the regular wreath product Gal ( L c / K c ) ≀ r Gal ( K c / F ) . Here Lc (resp. Kc ) denotes the Galois closure of L/F (resp. K/F). Similarly, we also construct an explicit embedding of the Galois group Gal ( L c / F ) into the smaller sized wreath product Gal ( L c / K ) ≀ Ω Gal ( K c / F ) , where Ω = Hom F ( K , K c ) is acted on by composition of automorphisms in Gal ( K c / F ) . Moreover, when L/K is a Kummer extension we prove a sharper embedding, that is, that Gal ( L c / F ) embeds into the wreath product Gal ( L / K ) ≀ Ω Gal ( K c / F ) . As corollaries we obtain embedding theorems when L/K is cyclic and when it is quadratic with char ( F ) ≠ 2 . We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry.