On a Problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers

TL;DR

This work addresses Erdős and Mirsky's problem on the ratio of divisor counts for consecutive integers by studying the closure $\mathcal{L}$ of $\

Abstract

Let $\mathcal{L}$ be the closure of the set of all real numbers $α$, such that there exist infinitely many integers $n$, such that $α=\log\frac{d(n+1)}{d(n)}$, where $d$ is the number of divisors of $n$. We give improved lower bounds for the density of $\mathcal{L}$.

Theorems & Definitions (6)

• Theorem 1.1: Hasanalizade
• Theorem 1.2
• Lemma 2.1
• Corollary 2.2
• proof
• Lemma 2.3: Macbeath