The irrationality of a prime factor series under a prime tuples conjecture

Abstract

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Assuming a suitably uniform version of the prime $k$-tuples conjecture, we show that the number \begin{align*} \sum_{n=1}^\infty \frac{\omega(n)}{2^n} \end{align*} is irrational. This settles (conditionally) a question of Erd\H{o}s.