Quantum mechanics and Riemann Hypothesis 

I was surprised when I first read about the connection between quantum mechanics and Riemann Hypothesis. Upon further reflection, their connections seem very natural. Here are some reasons.

First, the behavior of electrons is easier to study with complex functions. Electrons move around nucleus at extremely high speed. To be precise, an electron moves around the nucleus in a hydrogen atom at more than six thousand trillion rounds per second. That is much faster than our eyes, or any instrument, can follow. It is hopeless to study the movement of the electrons directly. However, electrons move in circles (roughly). Their frequency spectrum are easy to study. A Fourier transform turns time data into frequency data. This makes the behavior of electrons easier to study. Fourier transform turn real functions into complex functions. So quantum mechanics is based on complex functions. Riemann’s idea is to study the patterns of prime numbers, which are real numbers, from the perspective of complex numbers.

Second, the energy levels in quantum mechanics are discrete. The tool to study these energy levels is Schrodinger equation, whose solutions are continuous and differentiable. Prime numbers are discrete. The tool to study these prime numbers is Riemann Zeta function, which is an analytical complex function, differentiable everywhere except at one point.

Third, as energy level increases, discrete quantum states gradually morph into continuous states. Similarly, as the value of the imaginary part of the solutions of zeta function increases, the difference between the consecutive solutions approaches zero.

Over the last several decades, many papers uncovered the connections between quantum mechanics and Riemann Hypothesis. Why two theories have so much similarity?

My guess is that both theories are about periodic systems. Quantum mechanics is about periodic movements of the particles and their associated waves. All natural numbers are multiples of prime numbers. For example, all even numbers are multiples of two. Because of the fundamental property of periodicity, Fourier transform is important for both quantum mechanics and Riemann Hypothesis.