The shifted convolution of generalized divisor functions

TL;DR

This paper establishes an asymptotic formula for the shifted convolution of generalized divisor functions with a uniform error term, advancing previous results in the field.

Contribution

It provides a new asymptotic formula for the shifted convolution of divisor functions with improved error bounds and uniformity in the shift parameter.

Findings

Asymptotic formula for shifted convolution of divisor functions with power-saving error

Uniformity in the shift parameter

Improvement over previous results by Fouvry, Tenenbaum, and Drappeau

Abstract

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously by Fouvry and Tenenbaum and, more recently, by Drappeau.