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

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 !

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