Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series

TL;DR

This paper explores the connection between Rota's and Ramanujan's ideas, illustrating how probabilistic interpretations of Fourier series can shed light on additive number theory problems, bridging circle and sieve methods.

Contribution

It introduces a novel probabilistic perspective on Ramanujan-Fourier series, unifying circle and sieve methods in additive number theory.

Findings

Probabilistic interpretation enhances understanding of Ramanujan-Fourier series.

Circle and sieve methods are unified through probability and Fourier analysis.

New insights into additive number theory problems are provided.

Abstract

In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering.