Ratios of consecutive values of the divisor function

TL;DR

This paper proves that the ratios of consecutive divisor function values take on every positive rational number infinitely often, confirming a long-standing prediction by Erdős.

Contribution

It establishes that the sequence of ratios of consecutive divisor function values attains all positive rationals infinitely often, confirming Erdős's prediction.

Findings

Ratios of consecutive divisor function values are dense in positive rationals.

Every positive rational number appears infinitely many times as a ratio.

Confirms Erdős's prediction about the divisor function ratios.

Abstract

We show that the sequence of ratios $d(n+1) / d(n)$ of consecutive values of the divisor function attains every positive rational infinitely many times. This confirms a prediction of Erd\H{o}s.