On additive convolution sum of arithmetic functions and related questions

TL;DR

This paper extends Ingham's classical results on additive convolution sums of the divisor function, deriving new asymptotic formulas with error terms and exploring related sums involving functions with Ramanujan expansions.

Contribution

The paper generalizes Ingham's asymptotic formulas to sums with a variable upper limit and studies convolution sums of functions with Ramanujan expansions.

Findings

Derived asymptotic formula with error term for sum up to M

Extended analysis to convolution sums of functions with Ramanujan expansions

Provided insights into additive convolution sums of divisor functions

Abstract

Ingham studied two types of convolution sums of the divisor function, namely the shifted convolution sum $\sum_{n \le N} d(n) d(n+h)$ and the additive convolution sum $\sum_{n < N} d(n) d(N-n)$ for integers $N, h$ and derived their asymptotic formulas as $N \to \infty$. There have been numerous works extending Ingham's work on this convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result Ingham to derive an asymptotic formula with an error term of the sub-sum $\sum_{n < M} d(n) d(N-n)$ for an integer $M \le N$. Using this, we study the convolution sum $\sum_{n < M} f(n) g(N-n)$ for certain arithmetic functions $f$ and $g$ with absolutely convergent Ramanujan expansions, which in turm leads us to a well-established prediction of Ramanujan.