# Finite Ramanujan expansions and shifted convolution sums of arithmetical functions, II

## TL;DR

This paper advances the understanding of convolution sums of arithmetical functions by developing finite and shifted Ramanujan expansions, providing new tools for analyzing their asymptotic behavior and illustrating with classical examples.

## Contribution

It introduces a novel shifted Ramanujan expansion for convolution sums and compares it with existing finite expansions, deepening the theoretical framework for such sums.

## Key findings

- Development of a new shifted Ramanujan expansion for convolution sums
- Comparison between shifted and finite Ramanujan expansions
- Examples illustrating the application of these expansions in classical cases

## Abstract

We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\it shift} of the convolution sum. We deepen the study of finite Ramanujan expansions of general $f,g$ for the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum of $f$ and $g$, namely in terms of its shift $h$ and we compare this \lq \lq shifted Ramanujan expansion\rq \rq, with our previous finite expansions in terms of the $f$ and $g$ arguments. Last but not least, we give examples of such shift expansions, in classical literature, for the heuristic formul\ae.