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Multiplication Tables for Integers with Restricted Prime Factors

Jeremy Schlitt

Abstract

Let $Q$ be a set of primes with relative density $δ$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $δ\in (0,1]$. This generalizes the case $δ= 1$ from the 2008 work of Ford. We also show that there is a phase transition at the critical point $δ= 1/\log 4$, for which we explicitly determine the behaviour.

Multiplication Tables for Integers with Restricted Prime Factors

Abstract

Let be a set of primes with relative density . We count integers in with prime factors all in that also have a divisor in . We establish the order of magnitude for all . This generalizes the case from the 2008 work of Ford. We also show that there is a phase transition at the critical point , for which we explicitly determine the behaviour.
Paper Structure (8 sections, 24 theorems, 265 equations, 1 table)

This paper contains 8 sections, 24 theorems, 265 equations, 1 table.

Table of Contents

  1. Introduction
  2. Useful Lemmas
  3. The Lower Bound in Proposition 1.4
  4. The Upper Bound in Proposition 1.4
  5. Proof of \ref{['prop 1.5']}: Phase Transitions for a poisson-type sum
  6. Strong barriers for Generalized Smirnov Statistics
  7. Proof of Lower Bound Lemmas
  8. Proof of Upper Bound Lemmas

Key Result

Theorem 1.1

Suppose $100 \leq y \leq \sqrt{x}$. Then where

Theorems & Definitions (49)

  • Theorem 1.1: Ford ford2008distribution, 2008
  • Theorem 1.2
  • Corollary 1.3
  • Remark
  • Proposition 1.4
  • Proposition 1.5
  • proof : Proof of \ref{['mainthm']}
  • Lemma 2.1
  • proof
  • Lemma 2.2: koukoulopoulos2019distribution, Exercise 14.5
  • ...and 39 more