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Claim. The answer to Problem 69 is yes. Theorem 1.3 of Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 (v1 2025-12-01, v2 2026-04-25, 61 pages), states that

$$\sum_{n\ge1}\frac{\omega(n)}{2^n}=\sum_p\frac{1}{2^p-1}=0.5169428\ldots$$

is irrational. The $n=1$ term is zero, so the sum from $n\ge2$ in the site's statement is the same number; the identity comes from writing $\omega(n)=\sum_{p\mid n}1$ and summing the geometric series $\sum_{m\ge1}2^{-pm}$ for each prime $p$, an observation the site's remarks credit to Tao. In that form the theorem is the case of Problem 257 in which the infinite set is the set of primes. The source card is tao_2025_quantitative_correlations_problems_prime_factors_consecutive; Theorem 1.3 is quoted from arXiv v2; the corpus holds no review of its proof. The authors' main tool, as the card records, is a quantitative two-point correlation estimate for multiplicative functions (their Theorem 3.1) built on work of Pilatte, combined with probabilistic and circle-method arguments; the paper's other main results settle a conjecture of Erdős and Straus (Problem 248: infinitely many $n$ with $\omega(n+k)\ll k$ for every $k\ge1$) and prove, for all $x$ outside a set of logarithmic density zero, the asymptotic formula $(1+o(1))/(2\sqrt{\pi\log\log x})$ conjectured by Erdős, Pomerance and Sárközy for the proportion of $n\le x$ with $\omega(n)=\omega(n+1)$ (their Theorem 1.7, with analogues for $\Omega$ and $\tau$). This outline is a reading aid, not proof coverage.

Earlier work. Erdős proved in 1948 that $\sum_n\tau(n)/2^n$ is irrational and wrote that the analogous series for $\phi$, $\sigma$ and the number of prime factors seem to present difficulties (erdos_1948_arithmetical_properties_lambert_series). Pratt proved the irrationality of $\sum_n\omega(n)/t^n$ for every integer $t\ge2$ under a uniform quantitative prime tuples conjecture; that result has its own page, Pratt; the unconditional proof does not use it.

Acceptance. The reviewed evidence is the documented acceptance by the catalog erdosproblems.com: its curator, Thomas Bloom, credits the unconditional proof of irrationality to Tao and Teräväinen in the problem's remarks, and the page carries the label PROVED (last edited 2026-04-15); the problem has no forum comments and no proof claim. Crossref and the arXiv record list no journal publication so the claim carries no refereed evidence. The formal-conjectures statement file (the record link, pinned at its commit of 2026-10-06) tags, as of that commit, erdos_69 textbook and its specialization of Problem 257 research solved, both left as sorry, and records no formal proof. This corpus has not reproved the theorem and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited preprint alone.