A map of Ramanujan expansions

TL;DR

This paper provides a broad overview and new insights into Ramanujan expansions, focusing on finite and shift-Ramanujan expansions, while excluding extensive existing literature on additive and multiplicative functions.

Contribution

It introduces new types of Ramanujan expansions, specifically finite and shift-Ramanujan expansions, expanding the landscape of Ramanujan expansion research.

Findings

Introduction of finite Ramanujan expansions

Development of shift-Ramanujan expansions

Exclusion of extensive classical results on additive and multiplicative functions

Abstract

A map is a panorama in small scale. In this half-survey, half-research paper we give general results on Ramanujan expansions. We don't include the ocean of results from the literature on the two classes (see Schwarz-Spilker Book, also Lucht's survey for these) of additive and multiplicative functions while we include, say, the two new (not simply connected) lands of finite Ramanujan expansions (see my paper, with Murty & Saha) and of shift-Ramanujan expansions (see my subsequent paper, with Murty) .