Correlations of the divisor function

TL;DR

This paper investigates the linear correlations of the divisor function tau(n) using advanced methods, deriving asymptotic formulas for sums involving tau(n) and its shifted variants, extending understanding of divisor function behavior.

Contribution

It applies Green and Tao's methods to establish asymptotic formulas for correlations of the divisor function tau(n), a novel approach in this context.

Findings

Derived asymptotics for sums of tau(n) and its shifts

Extended understanding of divisor function correlations

Applied advanced analytic methods to number theory

Abstract

In this paper we study linear correlations of the divisor function tau(n) = sum_{d|n} 1 using methods developed by Green and Tao. For example, we obtain an asymptotic for sum_{n,d} tau(n) tau(n+d) ... tau(n+ (k-1)d).