On the Connection Between Riemann Hypothesis and a Special Class of Neural Networks

TL;DR

This paper explores a novel connection between the Riemann hypothesis and a special class of neural networks through an extension of the Nyman-Beurling criterion, offering new perspectives on this longstanding mathematical problem.

Contribution

It extends the Nyman-Beurling criterion by linking the Riemann hypothesis to a minimization problem involving neural networks, bridging number theory and machine learning.

Findings

Revisits and extends the Nyman-Beurling criterion

Establishes a connection between RH and neural network minimization

Provides an accessible introduction to RH for new audiences

Abstract

The Riemann hypothesis (RH) is a long-standing open problem in mathematics. It conjectures that non-trivial zeros of the zeta function all have real part equal to 1/2. The extent of the consequences of RH is far-reaching and touches a wide spectrum of topics including the distribution of prime numbers, the growth of arithmetic functions, the growth of Euler totient, etc. In this note, we revisit and extend an old analytic criterion of the RH known as the Nyman-Beurling criterion which connects the RH to a minimization problem that involves a special class of neural networks. This note is intended for an audience unfamiliar with RH. A gentle introduction to RH is provided.