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On the friable mean-value of the Erdős-Hooley Delta function

Bruno Martin, Gérald Tenenbaum, Julie Wetzer

Abstract

For integer $n$ and real $u$, define $Δ(n,u):= |\{d : d \mid n,\,{\rm e}^u <d\leqslant {\rm e}^{u+1} \}|$. Then, put $ Δ(n):=\max_{u\in{\mathbb R}} Δ(n,u).$ We provide uniform upper and lower bounds for the mean-value of $Δ(n)$ over friable integers, i.e. integers free of large prime factors.

On the friable mean-value of the Erdős-Hooley Delta function

Abstract

For integer and real , define . Then, put We provide uniform upper and lower bounds for the mean-value of over friable integers, i.e. integers free of large prime factors.
Paper Structure (5 sections, 2 theorems, 109 equations)

This paper contains 5 sections, 2 theorems, 109 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminary estimates
  3. Proof of Theorem \ref{['3est']}(i): lower bound
  4. Proof of Theorem \ref{['3est']}(i): upper bound
  5. Proof of Theorem \ref{['3est']}(ii)

Key Result

Theorem 1.1

(i) Let $\varepsilon>0$. For a suitable absolute constant $c>0$ and uniformly for $(x,y)\in H_\varepsilon$, we have (ii) For $2\leqslant y\leqslant x^{1/(2\log_2x\log_3x)}$, and with $\lambda:=y/\log x$, we have

Theorems & Definitions (3)

  • Theorem 1.1
  • Lemma 3.1
  • proof