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OP-9315OpenMathematics · general

Erdős #1049 · irrationality

Let t>1t>1 be a rational number. Is

∑n=1∞1tn−1=∑n=1∞τ(n)tn\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}

irrational, where τ(n)\tau(n) counts the divisors of nn?

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let t>1t>1 be a rational number. Is

∑n=1∞1tn−1=∑n=1∞τ(n)tn\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}

irrational, where τ(n)\tau(n) counts the divisors of nn?

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Background

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

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1049 link
  2. [2]Upstream Lean 4 statement for Erdős #1049 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1049 link

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