On the series of the reciprocals lcm's of sequences of positive integers: A curious interpretation

TL;DR

This paper investigates the average behavior of the smallest non-divisor elements in infinite sets of positive integers and relates it to the least common multiples of these sets, providing asymptotic results and discussing irrationality.

Contribution

It introduces a new limit formula connecting the average of smallest non-divisors to LCMs of initial segments of sets, with specific cases and irrationality considerations.

Findings

Derived a limit formula for the average of smallest non-divisors.

Provided asymptotic behavior for the set of all positive integers and primes.

Discussed the irrationality of the limit using Erdős's results.

Abstract

In this paper, we prove the following result: {quote} Let $\A$ be an infinite set of positive integers. For all positive integer $n$, let $\tau_n$ denote the smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to + \infty} \frac{1}{N} \sum_{n = 1}^{N} \tau_n = \sum_{n = 0}^{\infty} \frac{1}{\lcm\{a \in \A | a \leq n\}} .$${quote} In the two particular cases when $\A$ is the set of all positive integers and when $\A$ is the set of the prime numbers, we give a more precise result for the average asymptotic behavior of ${(\tau_n)}_n$. Furthermore, we discuss the irrationality of the limit of $\tau_n$ (in the average sense) by applying a result of Erd\H{o}s.