Analysing the Riemann Hypothesis and the approach to solve it through Zeta Function Analysis and Quantum-Inspired Random Matrix Theory

Author: Arnav Dhiman

The Riemann Hypothesis (RH) stands as one of the most famous unsolved problems in analytic number theory and mathematics as a whole. Its resolution promises profound implications and positive outcomes, particularly concerning our understanding of the distribution of prime numbers. Prime numbers are the fundamental building blocks of arithmetic and number theory, and proving the Riemann Hypothesis would provide a precise description of their distribution. The hypothesis posits that the "nontrivial" zeros of the Riemann zeta function all lie on a specific vertical line in the complex plane, known as the "critical line," which is located within a region called the "critical strip."

Furthermore, the spacing patterns of these nontrivial zeros exhibit a remarkable statistical similarity to the energy level spacings observed in chaotic quantum systems. This connection is described by Random Matrix Theory (RMT). In mathematics, the statistical distribution of zeta zeros strikingly matches the distribution of eigenvalues of certain random matrices. This observation has led to a profound idea: if the RH is true, there might exist a specific Hermitian operator whose eigenvalues correspond exactly to the imaginary parts of the zeta zeros. Since the eigenvalues of any Hermitian operator are always real numbers, the existence of such an operator would force all nontrivial zeta zeros to lie on the critical line, thereby proving the Riemann Hypothesis. This paper explores this promising approach, delving into the statistical similarities between the zeros of the Riemann zeta function and the eigenvalues of random matrices from the Gaussian Unitary Ensemble (GUE), and reviews the computational experiments that support this connection.

Introduction

Prime numbers have, since their discovery, been one of the most fascinating and challenging subjects in mathematics. Throughout history, mathematicians have relentlessly sought a concrete relationship or a general formula that could describe the seemingly random appearance of prime numbers. The reason primes are so important is that, according to the fundamental theorem of arithmetic, almost all whole numbers can be uniquely expressed as the product of prime numbers; consequently, they are often called the "building blocks of mathematics." Many of these attempts have indirectly paved the way for the formulation of the Riemann Hypothesis.

The Riemann Hypothesis is centered on the zeta function, a complex function analyzed by Bernhard Riemann.

The function has two types of zeros (points where the function's value is zero): a) Trivial Zeros b) Nontrivial Zeros

While the trivial zeros are well-understood and occur at the negative even integers, the nontrivial zeros are far more mysterious and hold the key to the hypothesis. According to the hypothesis, all nontrivial zeros lie on the "critical line," where the real part of the complex input is exactly 1/2. This line resides within the "critical strip," the region of the complex plane where the real part is between 0 and 1.

The Riemann Hypothesis is closely related to prime numbers and is based on an identity discovered by Euler that connects the zeta function to the prime numbers. Moreover, if we are able to prove that all the nontrivial zeroes lie on the critical line, we would be able to prove that the distribution of prime numbers, while not simple, is as regular and predictable as possible. Many mathematical theorems that are dependent on the assumption that the Riemann Hypothesis is “true” would finally be proven.

The fascinating and non-intuitive relation between quantum mechanics and the Riemann Hypothesis is that the way zeta zeros are distributed appears statistically identical to the distribution of energy levels in certain chaotic quantum systems, specifically those modeled by the Gaussian Unitary Ensemble (GUE) of random matrices. This paper aims to delve into this approach of using Random Matrix Statistics and the correlation between the GUE eigenvalue statistics and the spacing of the nontrivial zeros of the Riemann zeta function.

The Problem

The fundamental challenge in solving the Riemann Hypothesis is the lack of a definitive mathematical proof. While extensive computational searches have verified that the first several trillion nontrivial zeros do indeed lie on the critical line, this empirical evidence is not sufficient to constitute a conclusive mathematical proof. This problem can be explained further by understanding the Riemann hypothesis by building it upon the work of other mathematicians.

The Infinitude of Primes

The first question that needs to be answered is, “Are there an infinite number of primes?” The answer is yes, and this was famously proven by Euclid.

Theorem: There are infinitely many prime numbers.

Proof: Suppose, for contradiction, that there are only finitely many primes.

Let these primes be  p1,p2,p3.............pn

Now consider the number

N =  p1p2p3.............pn+1

When dividing N by any prime p(i) in our list, the remainder is 1.

Therefore, N is not divisible by any of the primes already in our list.

Thus, N must either be a prime itself or be divisible by some prime not in the list. In either case, we have found a prime not in our original collection, which is a contradiction. Therefore, there must be infinitely many prime numbers.

But Euclid's proof only showed that there are an infinite number of prime numbers; it didn't say anything about how these primes are spread out.

Euler's Product Formula and the Zeta Function

To analyse the distribution of primes, Leonhard Euler provided a crucial formula connecting the primes to the zeta function:

Euler’s product formula exposes the deep relationship between the prime numbers and the Riemann zeta function. This further helped us to get insights on the distribution of the prime numbers. However, there is still no proper formula governing the distribution of prime numbers.

The Prime Number Theorem

After Euler, the Prime Number Theorem (PNT) gave further insights into the distribution of primes, especially large ones.

Theorem: The prime number theorem gives an asymptotic form for the prime-counting function, π(x), which counts the number of primes less than or equal to some number x.

As x→∞:

π(x)~x/ln(x)​

This theorem tells us that primes become less frequent as numbers get bigger. But their “average density” is predictable: for a large number x, there is roughly one prime every ln(x) integers. Riemann saw that the zeros of the zeta function control how π(x) deviates from this average.

In other words, the Riemann Hypothesis could predict the error in the approximation given by the PNT, providing a much more precise account of when primes occur.

This implies that if RH is true, then the error term is as small as possible:

where Li(x) is the even more accurate logarithmic integral function.

The Formal Hypothesis

The Riemann zeta function is the function of the complex variable s, defined in the half-plane Re(s)>1 by the absolutely convergent series:

The Zeta Function is defined for the complex number s=σ+it. It can also be analytically continued to almost all complex numbers s except s=1, where it has a simple pole.

The zeros of the Zeta function include the following:

• Trivial Zeros: These occur at all negative even integers: s=−2,−4,−6,…

• Nontrivial Zeros: These are complex numbers s where ζ(s)=0 and 0<Re(s)<1.

The Riemann Hypothesis states that:

All nontrivial zeros of ζ(s) lie on the critical line where σ = 1/2

Mathematically,

The main problem with solving the Riemann hypothesis is that there are an infinite number of nontrivial zeroes of the Zeta function. Moreover, although over 10 trillion zeros have been computationally verified, a definitive mathematical proof that all zeros lie on the line remains elusive.

The Proposed Solution: Random Matrix Theory

While a definitive solution has not yet been found, a fascinating and relatively recent approach connects the problem to an unexpected field: Random Matrix Theory (RMT).

What is Random Matrix Theory?

In probability theory and mathematical physics, a random matrix is a matrix whose entries are random variables drawn from a specific probability distribution. Random Matrix Theory (RMT) is the study of the collective properties of these matrices, particularly their eigenvalues, as their size becomes very large. RMT employs powerful techniques such as mean-field theory and the replica method to compute statistical quantities like spectral densities and eigenvector correlations.

Montgomery’s Pair Correlation Conjecture

The relation between Random Matrix Theory and the Riemann Zeta function is given by Montgomery’s Pair Correlation Conjecture. A pair correlation function, in this context, studies the statistical distribution of the spacings between the nontrivial zeros of the zeta function.

Montgomery's pair correlation conjecture, published in 1973, asserts that the two-point correlation function for the normalized zeros of the Riemann zeta function on the critical line is:

Montgomery had conjectured that if you rescale the gaps between these zeros appropriately, their distribution should follow this formula.

The formula given above was a result from the Montgomery’s Pair Correlation Conjecture mathematical conjecture, which is given as:

Under the assumption that the Riemann hypothesis is true.

Let α≤β be fixed, then the conjecture states

It was then pointed out to him by physicist Freeman John that this formula is the same as the pair correlation function of random hermitian matrices. This is exactly the same formula as the pair spacing of eigenvalues in the Gaussian Unitary Ensemble (GUE) in random matrix theory.

The Gaussian Unitary ensemble is a set of hermitian matrices whose inputs are random values based on the gaussian normal distribution. Hermitian matrices always have real eigenvalues, so we can compare them directly to the real numbers.

Now when we find a similarity between the GUE and the correlation pair function of the Riemann Hypothesis it implies that since the spacing of the eigen values in the GUE are similar to the spacings of non trivial zeta zeroes then there may be a relation between them. Moreover, If RH is true, then maybe there exists a Hermitian operator whose eigenvalues correspond to the imaginary parts of zeta zeros and Since Hermitian operators have real eigenvalues, this would force all zeta zeros to lie on the critical line, this will prove RH. If we do find such a hermitian operator existing then RH will be proved.

In other words the Pair correlation of zeta zeros =Pair correlation of GUE eigenvalues.

To analyse this relation in the 1980s, mathematician Andrew Odlyzko set out to study the spacings between these zeroes through a computational approach and to test Montgomery's conjecture. This approach yielded the following results:

The following graphs show the similarity between the pair correlation of zeta zeroes and the eigenvalues of the GUE.

The scatter plot represents the observed density of spacings between zeros n for the Riemann Zeta function, with n ranging between the values written below the graph.

The solid line is the theoretical pair correlation function predicted by the Gaussian Unitary Ensemble (GUE) of random matrices.

These graphs show that there is an intricate relation between these correlation pairs from the random matrix theory and the non trivial zeroes of the Riemann Zeta function. The close match between the empirical data and the GUE prediction provides strong evidence for the connection between the zeros of the zeta function and eigenvalues of random Hermitian matrices, which was found by Montgomery as well, thus, supporting his decision.

As a part of this computation and result by Odlyzko, the Montgomery-Odlyzko law (which is a law in the sense of empirical observation instead of through mathematical proof) states that the distribution of the spacing between successive nontrivial zeros of the Riemann zeta function (suitably normalized) is statistically identical with the distribution of eigenvalue spacings in a Gaussian unitary ensemble.

The Gaussian Unitary Ensemble (GUE)

The Gaussian Unitary Ensemble is a set of Hermitian matrices whose entries are complex numbers with real and imaginary parts drawn independently from a standard Gaussian (normal) distribution. A key property of Hermitian matrices is that their eigenvalues are always real numbers. This allows for a direct comparison between the distribution of these eigenvalues and the distribution of the imaginary parts of the zeta zeros (which are also real numbers).

The statistical similarity between the eigenvalue spacings in the GUE and the spacings of the nontrivial zeta zeros suggests a deep, underlying connection. This leads to the Hilbert-Pólya conjecture, which states that if the RH is true, then there may exist a Hermitian operator whose eigenvalues correspond to the imaginary parts of the zeta zeros. Since the eigenvalues of a Hermitian operator are, by definition, real, the existence of such an operator would mathematically force all corresponding zeta zeros to lie on the critical line, thereby proving the RH. If such a Hermitian operator could be found, the RH would be proven.

In other words: Pair correlation of zeta zeros = Pair correlation of GUE eigenvalues.

Computational Evidence: The Work of Andrew Odlyzko

To analyse this relation, in the 1980s, mathematician Andrew Odlyzko set out to computationally test Montgomery's conjecture. His work produced striking numerical evidence supporting the connection.

The following graphs, based on Odlyzko's computations, visually demonstrate this remarkable similarity.

• The scatter plot represents the observed density of spacings between zeros γn​ for the Riemann Zeta function, computed over a very large range of zeros.

• The solid line is the theoretical pair correlation function predicted by the Gaussian Unitary Ensemble (GUE) of random matrices.

These graphs show that there is an intricate relation between these correlation pairs from random matrix theory and the nontrivial zeroes of the Riemann Zeta function. The close match between the empirical data and the GUE prediction provides strong evidence for the connection between the zeros of the zeta function and eigenvalues of random Hermitian matrices, thus supporting Montgomery's conjecture.

The Montgomery-Odlyzko Law

As a result of this computational work, the Montgomery-Odlyzko law (more of an empirically observed principle than a mathematically proven theorem) states that the distribution of the spacing between successive nontrivial zeros of the Riemann zeta function (suitably normalized) is statistically identical with the distribution of eigenvalue spacings in a Gaussian unitary ensemble.

Current Status of the Riemann Hypothesis

While the Riemann Hypothesis remains one of the most famous unsolved problems, we are at a critical juncture where our current mathematical tools may be insufficient to provide a proof.

• Unproven: The hypothesis remains an open problem. Despite immense effort, no accepted proof has been found.

• No Counterexamples: Extensive computational searches have not found a single nontrivial zero that lies off the critical line.

• Constraining Results: Many mathematical theorems have been proven that constrain where a counterexample could possibly lie, showing that if the RH is false, it must be false in a very specific and limited way.

• Promising Approaches: The connection to Random Matrix Theory and operator theory remains a highly active and promising area of research, but it has not yet yielded a complete proof.

• High Standard of Proof: Due to its importance, the mathematical community maintains an extremely high standard for any claimed proof, requiring rigorous and thorough verification by experts in the field.

Conclusion

The Riemann Hypothesis remains one of the most important unsolved problems in Mathematics and is subtly one of the most beautiful problems in the field. This hypothesis has the potential to explain the distribution of prime numbers, something crucial to understand not only for number theory but also for its applications in fields like cryptography and physics, as prime numbers are fundamental to these disciplines.

In conclusion, while significant progress has been made in understanding the problem, it is likely that new mathematical tools and conceptual breakthroughs will be required to finally solve it. Moreover, relatively recent approaches such as the Montgomery-Odlyzko law nudge us towards finding a different approach which is a non-intuitive yet compelling direction for future research. With ongoing research and the development of new mathematical tools, the global mathematical community remains hopeful that this century-old problem will eventually be solved.

Bibliography

• Berry, M. V., & Keating, J. P. (1999). The Riemann zeros and eigenvalue asymptotics. SIAM Review, 41(2), 236–266. [suspicious link removed]

• Conrey, J. B. (2003). The Riemann Hypothesis. Notices of the AMS, 50(3), 341–353. Link

• Edwards, H. M. (1974). Riemann’s Zeta Function. Dover Publications. Link

• Katz, N. M., & Sarnak, P. (1999). Random Matrices, Frobenius Eigenvalues, and Monodromy. American Mathematical Society. Link

• Mehta, M. L. (2004). Random Matrices (3rd ed.). Elsevier. Link

• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, 24, 181–193. Link

• Odlyzko, A. M. (1987). On the distribution of spacings between zeros of the zeta function. Mathematics of Computation, 48(177), 273–308. Link

• Riemann, B. (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, 671–680. Link

• Sarnak, P. (1999). Problems of the Millennium: The Riemann Hypothesis, Clay Mathematics Institute. Link

• Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-function (2nd ed.). Oxford University Press. Link