Finite Ramanujan expansions and shifted convolution sums of arithmetical functions

TL;DR

This paper introduces finite Ramanujan expansions for arithmetical functions and analyzes shifted convolution sums, providing asymptotic formulas with explicit error terms to advance understanding of their behavior.

Contribution

It extends previous work by defining finite Ramanujan expansions and applying them to derive asymptotic formulas for shifted convolution sums.

Findings

Derived explicit asymptotic formulas for convolution sums

Introduced finite Ramanujan expansions for arithmetical functions

Extended earlier theoretical frameworks

Abstract

For two arithmetical functions $f$ and $g$, we study the convolution sum of the form $\sum_{n \le N} f(n) g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup.