Physics of the Riemann Hypothesis (1101.3116v1)

Abstract: Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.

Physics of the Riemann Hypothesis

The paper "Physics of the Riemann Hypothesis" by Daniel Schumayer and David A.W. Hutchinson presents an extensive exploration of the connections between the Riemann zeta function and various areas of physics, emphasizing how physics might contribute to solving the long-standing Riemann Hypothesis. The authors explore the integral role of the Riemann zeta function across different branches of physics, stretching from classical mechanics to quantum chaos and condensed matter physics.

Key Aspects of the Riemann Zeta Function in Physics

• Historical Context and Mathematical Necessities:

The authors start with a historical perspective on number theory, outlining the importance of prime distribution, which is central to the Riemann zeta function. The function is defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$. Riemann hypothesized that all non-trivial zeros of $\zeta(s)$ lie on the "critical line" in the complex plane where $\Re(s) = 1/2$.

• Classical and Quantum Mechanics:

Classical mechanics models like billiards provide a basis to understand the notion of chaos and integrability. These concepts are fundamental in exploring the connection between microscopic classical paths and quantum spectra. A parallel is drawn with quantum mechanics via the analysis of trace formulae, emphasizing quantum systems that mirror the distribution of $\zeta(s)$ zeros.

• Quantum Chaos and Hilbert-Pólya Conjecture:

The authors address the Hilbert-Pólya conjecture, suggesting that the zeros of the Riemann zeta function could correspond to eigenvalues of an unknown self-adjoint operator, similar to a quantum Hamiltonian. There have been various quantum mechanical models proposed to substantiate this conjecture, including those simulating chaotic systems lacking time-reversal symmetry.

• Statistical Mechanics and Condensed Matter Physics:

Statistical mechanics is employed to explore Bose-Einstein condensation (BEC) and fluctuations, demonstrating dependency on spatial dimensions through the analytic properties of $\zeta(s)$. In condensed matter, the Riemann zeta and Epstein zeta functions connect to lattice properties and Madelung constants, linking with crystalline structures and binding energies.

Implications and Future Directions

The exploration of the Riemann zeta function suggests a strong interplay between mathematics and physics, encouraging cross-disciplinary approaches to longstanding problems. While the paper does not claim to present a solution to the Riemann Hypothesis, it highlights potential areas for fruitful research bridging quantum mechanics, statistical physics, and number theory.

The authors propose that further investigation into physical models corresponding to zeta function dynamics can stimulate advancements in both understanding the zeros and possibly leading to insights that could resolve the hypothesis. The discussion of random matrix theory also reinforces the idea of treating zeta zeros as quantum spectra, emphasizing symmetry and statistical distributions.

Concluding Remarks

This review paper serves as a comprehensive guide for researchers keen on exploring how physics can contribute to one of the most profound questions in mathematics—the Riemann Hypothesis. By exploring models and theories across different realms of physics, Schumayer and Hutchinson create a platform for future collaboration and discovery, underscoring the elegance with which physical principles can illuminate mathematical conjectures.