On additive convolution sum of arithmetic functions and related questions

Ingham studied two types of convolution sums of the divisor function, 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 result on the shifted convolution sum, but only little has been done towards the additive convolution sum. In this article, we extend the classical result of Ingham to derive an asymptotic formula with an error term of the sub-sum $\sum_{n < M} d(n) d(N-n)$ for certain integers $M \le N$. This involves careful choice of an applicable range of $M$. We also 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 turn leads us to a well-established prediction of Ramanujan.