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

Giovanni Coppola

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arXiv · 2502.14415

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

Giovanni Coppola

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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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Giovanni Coppola. 2025-02-20. Absolute convergence of Ramanujan expansions admits coefficients' coexistence of Ramanujan expansions. https://arxiv.org/abs/2502.14415

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