Erdős Problem #258

The problem

Let $a_1,a_2,\ldots$ be a sequence of positive integers with $a_n\to \infty$. Is

$$\sum_{n} \frac{\tau(n)}{a_1\cdots a_n}$$

irrational, where $\tau(n)$ is the number of divisors of $n$?

Statement quoted from erdosproblems.com/258, maintained by Thomas Bloom. Original source: [ErGr80,p.62][Er88c,p.103]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎