On Ramanujan smooth expansions for a general arithmetic function

TL;DR

This paper investigates Ramanujan smooth expansions for arithmetic functions, providing detailed local and global expansions for P-smooth and square-free arguments, advancing understanding of their structure and properties.

Contribution

It introduces the most general P-local Ramanujan expansions for P-smooth, square-free arguments and extends results to various subsets of arithmetic functions.

Findings

Derived explicit P-local expansions for P-smooth, square-free arguments

Extended Ramanujan expansions to specific subsets of arithmetic functions

Provided new insights into the structure of Ramanujan smooth expansions

Abstract

We study in detail the Ramanujan smooth expansions, for arithmetic functions; we start with the most general ones, for which we supply the "$P-$local expansions", for arguments with all prime-factors $p\le P$ (namely, $P-$smooth arguments), that are also square-free; then, we supply general results for interesting subsets of arithmetic functions, regarding both their $P-$local and (global) Ramanujan smooth expansions.