On additive convolution sum of arithmetic functions and related questions

TL;DR

This paper extends Ingham's additive convolution results for the divisor function to sub-sums and general Ramanujan-expandable arithmetic functions. It first derives an asymptotic for the sub-sum $\sum_{n\le M} d(n)d(N-n)$ with a main term proportional to $M$, a $\log^2$ factor involving $X=\sqrt{M(N-M)}$, and a power-saving error; it then develops a Ramanujan-expansion framework for general $f,g$, yielding a main term $M\sum_{r\ge 1} a_f(r)a_g(r)c_r(N)$ and explicit error terms depending on the decay exponent $\delta$. The applications to Ramanujan expansions of $\sigma$-types give explicit asymptotics for sums like $\sum_{n< M} \frac{\sigma_\alpha(n)}{n^\alpha} \frac{\sigma_\beta(N-n)}{(N-n)^\beta}$ and a corollary via partial summation for $\sum_{n<N} \sigma_\alpha(n)\sigma_\beta(N-n)$. This work sharpens the understanding of additive convolution sums and substantiates Ramanujan-based predictions for these arithmetic correlations.

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.

Theorems & Definitions (9)

• Theorem 1
• Remark 1
• Corollary 1
• Definition 1
• Theorem 2
• Lemma 1
• Remark 2
• Corollary 2
• Corollary 3