The Multiplication table for Smooth integers

TL;DR

This paper investigates the distribution of y-smooth integers within the multiplication table, extending the classical Erdős problem by focusing on integers with restricted prime factors.

Contribution

It introduces an analysis of y-smooth entries in the multiplication table, providing new insights into their quantity and distribution.

Findings

Quantifies the number of y-smooth integers in the multiplication table

Extends Erdős multiplication table problem to smooth integers

Provides asymptotic estimates for smooth entries

Abstract

The Erd\H{o}s multiplication table problem asks what is the number of distinct integers appearing in the $N\times N$ multiplication table. The order of magnitude of this quantity was determined by Ford in 2008. In this paper we study the number of $y-$smooth entries of the $N\times N$ multiplication table that is to say entries with no prime factors greater than $y$.