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  1. (, Journal of Number Theory)
    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. 
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    Free, publicly-accessible full text available November 1, 2026