Erdős Problem #1049 - Discussion thread

Let $t>1$ be a rational number. Is\[\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}\]irrational, where $\tau(n)$ counts the divisors of $n$?

#1049: [Er88c,p.102]

irrationality

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A conjecture of Chowla. Erdős [Er48] proved that this is true if $t\geq 2$ is an integer.

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This page was last edited 28 September 2025.

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1 comment on this problem

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Hi ! Chowla made a similar conjecture (on [Er48]). He conjectures that for all rational $|t|>1$, $g(1/t)=\sum_{n=1}^\infty \frac{\sin(n\pi/2)}{t^n-1}$ is irrational.
By using a similar method I use in my remark on problem [250], I think we can easily prove that $g(1/t)$ is transcendental for all algebric $|t|>1$. Did I make mistake ? Feel free to delete this comment if you find any mistake. Here’s my PDF : PDF
Thank you

RomanLeLan — 20:08 on 22 Oct 2025

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