New Perspective on the Riemann Hypothesis

Entitled “New Perspective on the Riemann Hypothesis”, the full version in PDF format is accessible in the “Free Books and Articles” section, here.

In about 10 pages (plus Python code, exercises and figures), this article constitutes a scratch course on the subject. It covers a large range of topics, both recent as well as unpublished, in a very compact style. Full of clickable references, the document covers the basics, offering a light reading experience. It also includes plenty of advanced, state-of-the-art material explained as simply as possible. Written by a machine learning professional working on experimental math, it is targeted to other machine learning professionals. Physicists, mathematicians, quants, statisticians and engineers will hopefully find this document easy to read, interesting, and opening up new research horizons. Exercise 8 is particularly intriguing, showing a potential new path to proving the Riemann Hypothesis.

Abstract

This tutorial provides a solid introduction to the Generalized Riemann Hypothesis and related functions, including Dirichlet series, Euler products, non-integer primes (Beurling primes), Dirichlet characters and Rademacher random multiplicative functions. The topic is usually explained in obscure jargon or inane generalities. To the contrary, this article will intrigue you with the beauty and power of this theory. The summary style is very compact, covering much more than traditionally taught in a first graduate course in analytic number theory. The choice of the topics is a little biased, with an emphasis on probabilistic models. My approach, discussing the “hole of the orbit” — called the eye of the Riemann zeta function in a previous article — is particularly intuitive.

The accompanying Python code covers a large class of interesting functions to allow you to perform as many different experiments as possible. If you are interested to know a lot more than the basics and possibly investigate this conjecture using machine learning techniques, this article is for you. The Python code also shows you how to produce beautiful videos of the various functions involved, in particular their orbits. This visual exploration shows that the Riemann zeta function, and a specific Dirichlet-L function (based on the non-trivial character modulo 4), behave very uniquely and similarly, explaining the connection between the Riemann and the Generalized Riemann Hypothesis, in pictures and videos rather than words.

Contents

Introduction

• Key concepts and terminology
• Orbits and holes
• Industrial applications

Euler products

• Finite Euler products
• Generalization using Dirichlet characters
• Infinite Euler products
• Special products
• Probabilistic properties and conjectures

Finite Dirichlet series and generalizations

• Finite Dirichlet series
• Non trivial cases with infinitely many primes and a hole
• Sums of two cubes, or cuban primes
• Primes associated to elliptic curves
• Analytic continuation, convergence, and functional equation
• Hybrid Dirichlet-Taylor series
• Riemann Hypothesis with cosines replaced by wavelets
• Riemann Hypothesis for Beurling primes
• Stochastic Euler products

Exercises

Python code

• Computing the orbit of various Dirichlet series
• Creating videos of the orbit