Fascinating Facts About Complex Random Variables and the Riemann Hypothesis

Despite my long statistical and machine learning career both in academia and in the industry, I never heard of complex random variables until recently, when I stumbled upon them by chance while working on some number theory problem. However, I learned that they are used in several applications, including signal processing, quadrature amplitude modulation, information theory and actuarial sciences. 

In this article, I provide a short overview of the topic, with application to understanding why the Riemann hypothesis (arguably the most famous unsolved mathematical conjecture of all times) might be true, using probabilistic arguments. Stat-of-the-art, recent developments about this conjecture are discussed in a way that most machine learning professionals can understand. The style of my presentation is very compact, with numerous references provided as needed. It is my hope that this will broaden the horizon of the reader, offering new modeling tools to her arsenal, and an off-the-beaten-path reading. The level of mathematics is rather simple and you need to know very little (if anything) about complex numbers. After all, these random variables can be understood as bivariate vectors (X, Y) with X representing the real part and Y the imaginary part. They are typically denoted as Z = X + iY, where the complex number i (whose square is equal to -1) is the imaginary unit. There are some subtle differences with bivariate real variables. The complex Gaussian variable is of course the most popular case.

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