# On Littlewood's proof of the prime number theorem

## TL;DR

This paper explores Littlewood's proof of the prime number theorem, extending it to establish an equivalence with the non-vanishing of the Riemann zeta-function on the critical line using almost periodic functions.

## Contribution

It introduces a new approach connecting Littlewood's proof with the non-vanishing of the zeta-function via almost periodic functions, providing a self-contained framework.

## Key findings

- Establishes an equivalence between the prime number theorem and zeta-function non-vanishing.
- Extends Littlewood's proof using almost periodic functions.
- Provides a self-contained proof approach.

## Abstract

In this note we examine Littlewood's proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the non-vanishing of Riemann's zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.