Absolute convergence of Ramanujan expansions admits coefficients' coexistence of Ramanujan expansions

TL;DR

This paper proves that every arithmetic function can be represented by infinitely many absolutely convergent Ramanujan expansions, demonstrating the non-uniqueness of the coefficients involved.

Contribution

It establishes the existence of infinitely many Ramanujan coefficients for any arithmetic function, highlighting the coexistence and non-uniqueness of such expansions.

Findings

Every arithmetic function has infinitely many Ramanujan coefficients.

All these coefficients produce absolutely convergent Ramanujan expansions.

The non-uniqueness of Ramanujan expansions is demonstrated for any given arithmetic function.

Abstract

In this self-contained short note, we prove that {\it every arithmetic function} $F$ {\it has infinitely many Ramanujan coefficients} $G$ {\it giving an absolutely convergent Ramanujan expansion for $F$}. This is "coefficients' coexistence": the non-uniqueness, once fixed any $F$, of these $G$. They are infinitely many !