An Overview of π and Euler Numbers, Including Their History, Relevance, and Current and Future Applications

Abstract

The mathematical constant $\pi$ and Euler numbers $E_n$ have long been of relevance in various branches of mathematics, particularly in number theory, combinatorics, numerical analysis, and mathematical physics. This review article introduces an exploration of their historical evolution, theoretical foundations, and recent advancements. We examine how $\pi$ and Euler numbers have facilitated advances in different scientific areas like quantum field theory and numerical algorithms. We introduce some emerging perspectives that highlight their interdisciplinary applications and potential future trajectories. Certainly, both numbers are of great relevance in current mathematical research, and their adaptability and ubiquity will continue to ensure their continuous appearance in future mathematical discoveries. The integration of $\pi$ and Euler numbers into advanced computational techniques and fields like artificial intelligence exemplifies their potential in driving mathematical innovation.

1. Introduction

Mathematical constants such as $\pi$ and Euler numbers have appeared in the development of various branches of mathematics. The constant $\pi$, defined as the ratio of a circle’s circumference C to its diameter d, is a fundamental element in real and complex analysis due to its transcendental nature [1]. Euler numbers, denoted by $E_n$ for $n\in {\mathbb{N}}_0$, arise in the study of alternating permutations and have important applications in combinatorics and number theory [2].

The transcendental nature of $\pi$ was established by Lindemann [3], who proved that $\pi$ is not a root of any non-zero polynomial equation with rational coefficients. This result has implications in number theory and algebraic geometry, particularly within the framework of Galois theory [4].

Euler numbers are significant in the Euler–Maclaurin formula, which connects discrete sums and continuous integrals, thus facilitating the approximation of sums through integrals and the analysis of asymptotic expansions [5]. Moreover, Euler numbers are connected to special functions and have applications in the study of Bernoulli numbers and Stirling numbers [6].

Recent advancements have seen $\pi$ and Euler numbers interfacing with contemporary mathematical disciplines such as quantum computing, information theory, and algebraic topology. The connections between these constants and modular forms, for example, have opened new avenues to understanding symmetries and invariants in higher-dimensional spaces [7].

This review article aims to present the historical evolution, current applications, and prospective future developments of $\pi$ and Euler numbers. In Section 2, we explore the historical evolution of these constants, tracing their origins and key milestones. Section 3 is concerned with the theoretical foundations and mathematical properties of $\pi$ and Euler numbers, including their analytical characteristics and interconnections. In Section 4, we present emerging perspectives along with recent theoretical developments and computational advances. Section 5 provides case studies of breakthroughs enabled by $\pi$ and Euler numbers, while Section 6 introduces potential areas of mathematical development in which $\pi$ and Euler numbers may have an impact. Eventually, Section 7 concludes the paper with the presentation of key points and a short reflection on the enduring significance of these constants.

2. Historical Evolution of $\mathit{\pi }$ and Euler Numbers

The mathematical constant $\pi$ and Euler numbers $E_n$ have a rich history contributing to the foundation and advancement of various mathematical disciplines. This section delineates the historical progression of these constants.

$\pi$, defined as the ratio of a circle’s circumference C to its diameter d,

$$\pi ={\displaystyle \frac{C}{d}},$$

(1)

has been recognized since ancient times. Early approximations of $\pi$ can be traced back to ancient civilizations. The Babylonians approximated $\pi$ as $3.125$ [8], while the Egyptians, as evidenced by the Rhind Papyrus, used ${\left({\displaystyle \frac{16}{9}}\right)}^2\approx 3.1605$ [9]. These approximations were derived from geometric methods involving the perimeters of polygons inscribed within and circumscribed around a unit circle.

Archimedes of Syracuse made significant strides in approximating $\pi$ through the method of exhaustion, a precursor to integral calculus. Based on the idea of inscribing and circumscribing regular polygons with an increasing number of sides within a unit circle, Archimedes established the following:

$${\displaystyle \frac{223}{71}}\approx 3.1408<\pi <{\displaystyle \frac{22}{7}}\approx 3.1429.$$

(2)

This method involves calculating the perimeters $P_n$ of regular n-gons inscribed in a unit circle, where n is even. The perimeter is given by

$$P_n=n·sin\left({\displaystyle \frac{\pi }{n}}\right).$$

(3)

As n increases, $P_n$ converges to the circumference $C=2\pi$, providing a systematic approach to approximate $\pi$ with greater precision.

The quest for more accurate values of $\pi$ continued through the centuries. In the 17th century, infinite-series representations of $\pi$ began to emerge, laying the groundwork for its analysis within calculus. Gottfried Wilhelm Leibniz introduced the Leibniz formula for $\pi$:

$$\pi =4\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k}{2k+1}},$$

(4)

which converges conditionally to $\pi$. This series is a special case of the Taylor series expansion for the arctangent function:

$$arctan\left(x\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{x}^{2k+1}}{2k+1}},\left|x\right|\le 1.$$

(5)

Evaluating $x=1$ yields the Leibniz formula.

In the 18th century, Johann Bernoulli introduced more efficient series for calculating $\pi$. The development of continued fractions and iterative algorithms further refined methods for determining $\pi$ to an arbitrary number of decimal places, which became of high interest for advancements in computational mathematics and engineering applications [10].

As previously mentioned, the transcendental nature of $\pi$ was conclusively established by Ferdinand von Lindemann in 1882 [3]. This proof relies on the Hermite–Lindemann theorem, which asserts that if $\alpha$ is a non-zero algebraic number, then $e^{\alpha }$ is transcendental. Applying this theorem, since $e^{i\pi }=-1$ and $-1$ is algebraic, it follows that $i\pi$ must be transcendental. Consequently, $\pi$ itself is transcendental. This result has an important consequence, namely, the impossibility of squaring the circle using a finite number of steps with a compass and straightedge (refer to [1] for additional information).

Now, let us focus on Euler numbers, $E_n$, and particularly on the enumeration of alternating permutations. An alternating permutation of order n is a permutation $\sigma \in S_n$, the symmetric group on n elements, satisfying

$$\sigma \left(1\right)<\sigma \left(2\right)>\sigma \left(3\right)<\sigma \left(4\right)>\cdots .$$

Indeed, the number of such permutations is given by the Euler number $E_n$. These numbers can be generated using the generating function

$$sec\left(x\right)+tan\left(x\right)=\sum _{n=0}^{\infty }{E}_n{\displaystyle \frac{x^n}{n!}},\left|x\right|<{\displaystyle \frac{\pi }{2}}.$$

(6)

This function facilitates the derivation of various properties of Euler numbers, such as their recurrence relations and connections to other mathematical constructs.

In addition, Euler numbers satisfy the recurrence relation

$$E_{n+1}=-\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)E_k,n\ge 0,$$

(7)

with the initial condition $E_0=1$. The sequence is $E_0=1$, $E_1=0$, $E_2=-1$, $E_3=0$, $E_4=5$, and so forth (the reader is referred to [6,11] for additional details).

Euler numbers also appear in the Euler–Maclaurin summation formula, which connects discrete sums with continuous integrals. The formula is expressed as

$$\sum _{k=a}^{b}f\left(k\right)={\int }_a^{b}f\left(x\right)dx+{\displaystyle \frac{f\left(b\right)+f\left(a\right)}{2}}+\sum _{k=1}^m{\displaystyle \frac{B_{2k}}{\left(2k\right)!}}\left(f^{(2k-1)}\left(b\right)-f^{(2k-1)}\left(a\right)\right)+R_m,$$

(8)

where $B_{2k}$ is a Bernoulli number, and $R_m$ is the remainder term [5]. In this context, Euler numbers contribute to the coefficients in the asymptotic expansion of the remainder term, enabling the approximation of sums by integrals with controlled error terms.

Throughout the 19th and 20th centuries, Euler numbers found applications beyond combinatorics and extended into numerical analysis, approximation theory, and the study of special functions (as we already mentioned in the introduction, there exist connections with Bernoulli numbers and Stirling numbers of the first kind; refer to [12] for additional details).

We now explore some connections between $\pi$ and Euler numbers. As briefly mentioned before, they appear in the theory of modular forms, which are complex functions exhibiting specific transformation properties under the action of the modular group. Modular forms are relevant for the number theory, particularly in the proof of Fermat’s Last Theorem [13]. The symmetries and invariants captured by modular forms often involve $\pi$ through their Fourier coefficients, which can be expressed using Euler numbers in certain contexts.

In algebraic topology and differential geometry, $\pi$ appears in the characterization of manifolds and curvature, while Euler numbers contribute to topological invariants such as the Euler characteristic. The Euler characteristic, defined for a compact even-dimensional manifold M as

$$\chi \left(M\right)=\sum _{k=0}^n{(-1)}^{k}dim{H}_k(M;\mathbb{Q}),$$

(9)

where $H_k(M;\mathbb{Q})$ denotes the k-th homology group with rational coefficients, encapsulates topological information about M. Both $\pi$ and Euler numbers appear in the formulation and resolution of different problems within these advanced mathematical areas.

3. Theoretical Foundations and Mathematical Properties

In real and complex analyses, $\pi$ appears ubiquitously in the study of periodic functions, Fourier series, and integral transforms. A notable example is the Fourier series representation of periodic functions, where $\pi$ serves as a frequency parameter. For a function $f\in L^2(-\pi ,\pi )$, the Fourier series expansion is given by

$$f\left(x\right)={\displaystyle \frac{a_0}{2}}+\sum _{n=1}^{\infty }\left(a_{n}cos\left(nx\right)+b_{n}sin\left(nx\right)\right),$$

(10)

where the coefficients $a_n$ and $b_n$ are determined by

$$\begin{array}{cc}\hfill a_n& ={\displaystyle \frac{1}{\pi }}{\int }_{-\pi }^{\pi }f\left(x\right)cos\left(nx\right)dx,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill b_n& ={\displaystyle \frac{1}{\pi }}{\int }_{-\pi }^{\pi }f\left(x\right)sin\left(nx\right)dx.\hfill \end{array}$$

(12)

Here, $\pi$ facilitates the orthogonality of the sine and cosine functions. This property is well known in signal processing, differential equations, and various applied mathematics fields [14].

Beyond Fourier analysis, $\pi$ is important in the theory of continued fractions, which provide representations of real numbers through infinite sequences of integer quotients. The continued fraction expansion of $\pi$ is of particular interest in number theory due to its non-repeating and non-terminating nature (certainly this is a reflection of the transcendental property of $\pi$). An example of a continued fraction for $\pi$ is

$$\pi =3+{\displaystyle \frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\cdots }}}}},$$

(13)

which clearly introduces the complexity of $\pi$’s numerical structure [10].

Integral representations of $\pi$ further highlight its analytical richness. The Gaussian integral in probability theory and quantum mechanics is expressed as

$${\int }_{-\infty }^{\infty }{e}^{-x^2}dx=\sqrt{\pi }.$$

(14)

This integral connects $\pi$ to the normalization constant of the normal distribution as it typically appears in mechanics and thermodynamics [15].

Let us now turn our attention to Euler numbers $E_n$. The generating function for Euler numbers is given in Equation (6), and this generating function is analytic within its radius of convergence and serves as a foundation for deriving properties such as recurrence relations and generating higher-order Euler numbers. Euler numbers also appear in the study of orthogonal polynomials and special functions. Specifically, they appear in the expansion of secant and tangent functions, which are closely related to Euler polynomials. Euler polynomials $E_n\left(x\right)$ are defined by the generating function

$${\displaystyle \frac{2e^{xt}}{e^t+1}}=\sum _{n=0}^{\infty }{E}_n\left(x\right){\displaystyle \frac{t^n}{n!}},\left|t\right|<\pi .$$

(15)

These polynomials satisfy various orthogonality relations and differential equations, making them important in approximation theory and mathematical physics [12].

In number theory, both $\pi$ and Euler numbers contribute to the understanding of special functions and zeta functions. The Riemann zeta function, defined for $\Re \left(s\right)>1$ by

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}},$$

(16)

intersects with Euler numbers through its functional equations and connections to Bernoulli numbers. In addition, Euler numbers appear in the Laurent series expansion of the zeta function at specific points, linking them to the distribution of prime numbers and the analytical continuation of $\zeta \left(s\right)$ [16].

In mathematical physics, $\pi$ appears in various contexts. For instance, in the path integral formulation of quantum mechanics, integrals involving $\pi$ are ubiquitous due to their connection with Gaussian integrals (refer to Expression (14)) and normalization constants [15]. Euler numbers also emerge in the study of Feynman diagrams and perturbative expansions, where they contribute to the enumeration of certain classes of diagrams and interactions.

In algebraic geometry, $\pi$ appears in the study of complex manifolds and their curvature properties, while Euler numbers contribute to the classification of vector bundles and characteristic classes [17].

4. Emerging Perspectives

This section elucidates some theoretical breakthroughs and prospective future trajectories that are poised to further our understanding and utilization of $\pi$ and Euler numbers.

Recent work by Zudilin [18] explored linear independence measures involving $\pi$ and other fundamental constants, contributing to the broader framework of irrationality and transcendence proofs. These kinds of studies aim to establish new relationships between $\pi$ and other transcendental numbers, potentially uncovering novel algebraic structures and symmetries within number theory.

In the area of combinatorics, Knuth and Wilf [19] introduced generalized generating functions that extend the classical definitions of Euler numbers as provided in Expressions (6) and (7), facilitating their application in more complex combinatorial structures such as hypergeometric series and partition functions. These generalized frameworks allow for the enumeration of higher-dimensional alternating permutations and other combinatorial objects.

In computational mathematics, the development of high-performance algorithms, such as the Gauss–Legendre and the Chudnovsky algorithms, has enabled the computation of $\pi$ to billions of decimal places with higher speed and accuracy [10]. Moreover, the integration of $\pi$ and Euler numbers into quantum computing frameworks has opened new areas for exploration. Quantum algorithms, which exploit the principles of superposition and entanglement, utilize $\pi$ in the formulation of quantum gates and error correction protocols [20]. Euler numbers, with their combinatorial properties, are employed in optimizing quantum circuits and analyzing quantum entanglement patterns, thus contributing to the development of more efficient and robust quantum computing systems [21].

In cryptography, the transcendental nature of $\pi$ appears in the development of secure encryption algorithms. Patel and Gupta [22] investigated the use of $\pi$ in generating cryptographic keys with high entropy to improve the security of digital communication systems. Additionally, Euler numbers are utilized in the construction of cryptographic protocols that rely on combinatorial hardness assumptions, providing robust frameworks for secure data transmission [23].

Future research trajectories anticipate the exploration of $\pi$ within non-commutative geometry, a branch of mathematics that extends classical geometric concepts to quantum spaces [24]. This exploration aims to redefine geometric structures and curvature properties in higher-dimensional and non-commutative contexts, potentially leading to new ideas in both mathematics and theoretical physics. Concurrently, Euler numbers can be used in higher-dimensional combinatorics and algebraic topology as well as in the theory of topological invariants. This application should extend their applicability to the classification of complex manifolds and fiber bundles [25].

The synergy between $\pi$ and Euler numbers in machine learning and artificial intelligence represents another promising frontier. According to [26], the mathematical properties of these constants are of interest for developing advanced algorithms for pattern recognition, data compression, and optimization. The application of AI-driven research tools to discover new mathematical theorems involving $\pi$ and Euler numbers is anticipated to accelerate mathematical innovation, and this enables the automated exploration of complex mathematical problems [27].

5. Case Studies

This section presents case studies that exemplify how $\pi$ and Euler numbers have enabled advancements in number theory, combinatorics, numerical analysis, and mathematical physics.

A notable case study is the application of $\pi$ in the development of the Fast Fourier Transform (FFT) algorithm. The FFT algorithm efficiently computes the discrete Fourier transform (DFT) of a sequence, reducing the computational complexity from $O\left(N^2\right)$ to $O(NlogN)$ [28]. The DFT is defined for a sequence of complex numbers ${\left\{x_n\right\}}_{n=0}^{N-1}$ as

$$X_k=\sum _{n=0}^{N-1}{x}_n{e}^{-2\pi ikn/N},k=0,1,\cdots ,N-1,$$

(17)

where $\pi$ appears in the exponential term, ensuring the periodicity and orthogonality of the basis functions. The FFT algorithm considers the periodic properties governed by $\pi$ to decompose signals into their frequency components, which is important in fields like digital signal processing, image analysis, and telecommunications [29].

Euler numbers appear in the analysis of finite difference schemes used in numerical solutions of differential equations. Specifically, they appear in the error analysis of these schemes, where they help to quantify the truncation errors associated with approximating derivatives. For example, we may consider a finite difference approximation of the second derivative:

$$f^{″}\left(x\right)\approx {\displaystyle \frac{f(x+h)-2f\left(x\right)+f(x-h)}{h^2}},$$

(18)

where h is the step size. In this context, the Euler–Maclaurin summation formula, which involves Euler numbers, provides a framework for analyzing the convergence and precision of such numerical methods by relating discrete sums to continuous integrals with controlled error terms [30]. This particular case shows the importance of Euler numbers in ensuring the reliability and precision of numerical simulations in scientific computing and engineering.

Another particular and relevant case is provided in quantum field theory (QFT). The path integral formulation of QFT for a scalar field $\varphi \left(x\right)$ is expressed as

$$Z=\int \mathcal{D}\varphi e^{iS\left[\varphi \right]/\hslash },$$

(19)

where $S\left[\varphi \right]$ is the action functional and $\mathcal{D}\varphi$ denotes the functional measure over all field configurations. The normalization factors involving $\pi$ ensure the convergence of these integrals in the perturbative expansion of QFT [31].

Another particular case is provided in quantum electrodynamics (QED), whereby the number of loop diagrams at a given order can be related to Euler numbers through generating functions and combinatorial identities [32]. This combinatorial aspect allows us to organize and calculate contributions to physical quantities in QFT, such as scattering amplitudes and correlation functions.

Another significant case study involves the use of $\pi$ in the derivation of the Euler–Bernoulli beam theory in structural engineering. The Euler–Bernoulli beam equation describes the relationship between the beam’s deflection $w\left(x\right)$ and the applied load $q\left(x\right)$:

$$EI{\displaystyle \frac{d^{4}w\left(x\right)}{d{x}^4}}=q\left(x\right),$$

(20)

where E is Young’s modulus, and I is the second moment of the area of the beam’s cross-section. The constant $\pi$ appears in the calculation of I for circular cross-sections:

$$I={\displaystyle \frac{\pi r^4}{4}},$$

(21)

where r is the radius of the circular cross-section. On this occasion, $\pi$ appears in the determination of the bending stiffness of beams, which is critical for designing stable and resilient structures in civil engineering [33].

Euler numbers also emerge in the study of random matrix theory, which has applications in number theory, statistical mechanics, and quantum chaos. In random matrix theory, the distribution of eigenvalues of large random matrices is analyzed, and Euler numbers appear in the asymptotic expansions of certain correlation functions. For example, the probability density function for the spacing between adjacent eigenvalues in the Gaussian Unitary Ensemble (GUE) involves Euler numbers through their connection with orthogonal polynomials and combinatorial structures [34]. This connection facilitates the understanding of complex systems and the statistical properties of spectra in various physical and mathematical contexts.

Although we previously mentioned the Riemann zeta function $\zeta \left(s\right)$, it is important to consider further details given its relevance as a case study. Particularly, the functional equation of the Riemann zeta function involves $\pi$ through the gamma function $\mathsf{\Gamma }\left(s\right)$:

$$\zeta \left(s\right)=2^s{\pi }^{s-1}sin\left({\displaystyle \frac{\pi s}{2}}\right)\mathsf{\Gamma }(1-s)\zeta (1-s).$$

(22)

This equation connects values of $\zeta \left(s\right)$ at s and $1-s$, and $\pi$ ensures the symmetry and analytic continuation of $\zeta \left(s\right)$ across the complex plane [16].

Another interesting case that warrants further research (although it has previously been commented on generally) concerns the theory of L-functions in the study of modular forms. In particular, Euler numbers appear in the coefficients of the Fourier expansions associated with the formulation of L-functions, which generalize the Riemann zeta function and are important for many conjectures and theorems in number theory [7]. The connection between Euler numbers and modular forms enhances the analytical tools available for probing the properties of L-functions and their zeros, which are deeply connected to the distribution of prime numbers.

Euler numbers have also been applied in the optimization of algorithms. Specifically, they assist in the enumeration and analysis of permutation patterns, which appear in fields such as cryptography, bioinformatics, and computer science. Considering the combinatorial properties of Euler numbers, it is possible to design algorithms that efficiently solve complex enumeration problems and optimize search strategies [35]. This application shows the versatility of Euler numbers in enhancing algorithmic performance and solving practical problems in various technological domains.

6. Future Directions of $\mathit{\pi }$ and Euler Numbers

Although $\pi$ and Euler numbers have been studied extensively, they continue to surface in unexpected ways within cutting-edge mathematical research. In this section, unlike in previous sections, we will focus on analyzing the potential areas of mathematics (mainly pure mathematics) where these numbers are significant. In doing so, we aim to introduce the question of whether the majority of mathematics concerning $\pi$ and Euler numbers has already been already postulated at least on a general basis. The truth is that this is not the case, and advances in mathematics demonstrate how these numbers are integral to exploration—either as epistemological elements or as instrumental means to construct more elegant and simpler theories within mathematics.

Let us consider the field of quantum topology, where $\pi$ emerges in the study of knot invariants and three-dimensional manifolds. Specifically, the Chern–Simons theory employs $\pi$ in its action functional, which is important for the quantization of gauge fields on 3-manifolds [36]. The Chern–Simons action is given by

$$S_{\mathrm{CS}}\left(A\right)={\displaystyle \frac{k}{4\pi }}{\int }_{M}Tr\left(A\wedge dA+{\displaystyle \frac{2}{3}}A\wedge A\wedge A\right),$$

(23)

where A is a connection on a principal G-bundle over a 3-manifold M, and k is the coupling constant. The appearance of $\pi$ in this context is not merely a normalization choice; it ensures the correct quantization conditions for the theory.

Recent work has connected Chern–Simons theory to knot homologies, such as Khovanov homology, and to the categorification of quantum link invariants [37]. These developments hint at deeper structures where $\pi$ could be connected with category theory and higher-dimensional algebra, opening avenues for new ideas in invariants of knots and 3-manifolds.

Another area of interest for Euler numbers is enumerative geometry, particularly within the framework of mirror symmetry. In this context, the Euler characteristic of a Calabi–Yau manifold is a quantity influencing the counting of rational curves and the calculation of Gromov–Witten invariants [38]. Mirror symmetry predicts that the complex geometry of a Calabi–Yau manifold is reflected in the symplectic geometry of its mirror partner, and Euler numbers appears in this duality.

As a short example, consider the prediction of instanton numbers $n_d$ for a quintic threefold in ${\mathbb{P}}^4$. The generating function for these numbers involves the Euler characteristic $\chi$ of the manifold:

$$F\left(t\right)=\sum _{d=1}^{\infty }{n}_d{q}^d,q=e^{2\pi it},$$

(24)

where t is the complexified Kähler parameter. Future research may consider generalizing these results to higher-dimensional Calabi–Yau manifolds and understanding the relevance of Euler numbers in the enumerative geometry of Gopakumar–Vafa invariants [39].

We now introduce some ideas beyond the transcendental nature of $\pi$. In the context of the Langlands program, $\pi$ appears in the functional equations of automorphic L-functions and in the normalization of Haar measures on adele groups [40]. The Langlands correspondence seeks to relate Galois representations and automorphic forms, and $\pi$ is subtly woven into this grand unifying theory. Emerging research is exploring the intersection of $\pi$ in p-adic Langlands correspondence and in the study of motivic L-functions. There is potential for $\pi$ to feature in new reciprocity laws and to contribute to the understanding of special values of L-functions, which are admitted to be part of important conjectures like the Bloch–Kato conjecture [41].

Euler numbers continue to inspire discoveries in combinatorics and special functions. Their appearance in Faulhaber’s formula for sums of powers illustrates deep connections between discrete and continuous mathematics:

$$\sum _{k=1}^n{k}^m={\displaystyle \frac{1}{m+1}}\sum _{j=0}^m{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{j}}\right)B_j{n}^{m+1-j},$$

(25)

where $B_j$ represents Bernoulli numbers. Generalizations involving Euler numbers could lead to new summation formulas and identities.

Recent studies have introduced generalized Euler numbers and their q-analogs, which have applications in partition theory and q-series [42]. These generalizations may unlock new relationships between combinatorial structures and special functions, such as basic hypergeometric functions, with potential implications for number theory and mathematical physics.

Random matrix theory (RMT) has provided new ideas concerning the distribution of the zeros of the Riemann zeta function, particularly those lying on the critical line. We recall that the Riemann zeta function, $\zeta \left(s\right)$, is a complex function defined for complex numbers $s=\sigma +it$, where $\sigma$ represents the real part and t the imaginary part of s. The nontrivial zeros of $\zeta \left(s\right)$ are conjectured by the Riemann Hypothesis to lie on the critical line where $\sigma ={\displaystyle \frac{1}{2}}$.

RMT studies the properties of matrices with randomly distributed entries and is particularly useful in understanding the statistical behavior of eigenvalues of large random matrices. Eigenvalues are special numbers that satisfy the equation $A\mathbf{v}=\lambda \mathbf{v}$, where A is a matrix, $\lambda$ is an eigenvalue, and $\mathbf{v}$ is the corresponding eigenvector. In the context of RMT, the spacing between consecutive eigenvalues, when ordered numerically, is of particular interest. This spacing is often normalized to account for the average density of eigenvalues, resulting in a normalized spacing denoted by s.

The probability distribution function $P\left(s\right)$ describes the likelihood of a specific normalized spacing s between eigenvalues. In RMT, this function is given by

$$P\left(s\right)={\displaystyle \frac{\pi }{2}}sexp\left(-{\displaystyle \frac{\pi }{4}}{s}^2\right).$$

(26)

This equation indicates that small spacings (s close to 0) are less likely, while spacings around $s=1$ are more probable. As s increases, the probability decreases exponentially, reflecting the repulsion between eigenvalues. The Montgomery–Odlyzko law posits a connection between RMT and the distribution of the nontrivial zeros of the Riemann zeta function. Specifically, it suggests that the statistical properties of these zeros on the critical line are similar to the eigenvalues of large random matrices. This connection implies that techniques from quantum chaos, which studies the behavior of quantum systems that are classically chaotic, can be applied to understand the distribution of prime numbers.

The presence of $\pi$ in the probability distribution function $P\left(s\right)$ highlights a link between quantum chaos and prime number theory. Indeed, quantum chaos provides a framework for analyzing systems with complex, chaotic behavior using the principles of quantum mechanics. On the other hand, prime number theory deals with the properties and distribution of prime numbers, which are the building blocks of arithmetic. The possible connections between these fields suggests that the inherent properties of $\pi$ and the structures revealed by RMT could eventually introduce new ideas for further understating the long-standing Riemann Hypothesis.

7. Conclusions

The constant $\pi$ and Euler numbers $E_n$ are important pillars in mathematics, as they have shaped and are continuously shaping both historical and modern theories. In this review article, we have presented their impact, which ranges from early geometry and number theory to advanced fields like quantum computing and machine learning. Theoretical developments and computational techniques continue to reveal their deep interconnections and applications. The future of research in this area is promising, with challenging areas likely to be explored, for example the influence of $\pi$ in non-commutative geometry as well as the extension of Euler numbers into higher-dimensional combinatorics (among many others).