Erdős Problem #259

Is the sum\[\sum_{n} \mu(n)^2\frac{n}{2^n}\]irrational?

#259: [ErGr80,p.62][Er81h,p.180][Er81l][Er88c,p.103]

irrationality

In [Er88c] Erdős makes the stronger conjecture that any infinite subseries over squarefree numbers is also irrational.

This strong conjecture is true, and was proved by Chen and Ruzsa [ChRu99].

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This page was last edited 19 October 2025.

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Formalised statement? Yes
Related OEIS sequences: A371134

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Additional thanks to: Boris Alexeev, Dustin Mixon, Terence Tao, and Desmond Weisenberg

When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #259, https://www.erdosproblems.com/259, accessed 2026-02-13