The Differences Between Consecutive Smooth Numbers

TL;DR

This paper investigates the distribution of smooth numbers, demonstrating that large gaps between them are rare, using a novel mean value bound for a specific Dirichlet polynomial as a key tool.

Contribution

Introduces a new mean value bound for Dirichlet polynomials to analyze the gaps between smooth numbers, revealing their infrequency.

Findings

Large gaps between smooth numbers are infrequent

New mean value bound for Dirichlet polynomial established

Enhanced understanding of smooth number distribution

Abstract

We show that large gaps between smooth numbers are infrequent. The key new tool is a novel mean value bound for a special type of Dirichlet polynomial.