On the greatest common divisor of $n$ and the $n$th Fibonacci number, II

TL;DR

This paper investigates the distribution of the set of integers formed by the gcd of n and the nth Fibonacci number, providing an upper bound on their density within large intervals.

Contribution

It establishes a refined upper bound on the count of such gcd values up to x, improving understanding of their distribution.

Findings

The set of gcd values has density zero.

The number of such gcds up to x is bounded by x times a slowly growing logarithmic factor.

Provides a quantitative estimate for the distribution of gcds with Fibonacci numbers.

Abstract

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. Leonetti and Sanna proved that $\mathcal{A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that \begin{equation*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{equation*} for all sufficiently large $x$.