Erdős Problem #259 - Discussion thread

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.

External data from the database - you can help update this
Formalised statement? Yes
Related OEIS sequences: A371134

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 2025-11-15

1 comment on this problem

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I got an AI to compute this sequence (among others), searched for it in the OEIS, and found a hit with A371134. That page links to a positive solution to this problem at

Yong-Gao Chen and Imre Z. Ruzsa, On the irrationality of certain series, Periodica Mathematica Hungarica, Vol. 38, No. 1 (1999), pp. 31-37.

So the erdosproblems.com / OEIS linkage project is bearing fruit!

(The site has been updated to address this comment.)

TerenceTao — 16:09 on 02 Sep 2025

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