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

TL;DR

Under a conjecture related to prime tuples, the paper proves that a specific infinite series involving the number of prime factors is irrational, addressing a question posed by Erdős.

Contribution

The paper demonstrates the irrationality of an infinite series involving prime factors under a conjecture about prime tuples, providing a conditional answer to Erdős's question.

Findings

Proves the series is irrational assuming the prime k-tuples conjecture.

Links the irrationality to a deep conjecture in prime number theory.

Addresses a longstanding question of Erdős conditionally.

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.