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The number of preimages of iterates of $φ$ and $σ$

Agbolade Patrick Akande

Abstract

Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function $φ$ and the sum-of-divisors functions $σ$. In this paper, we will extend the upper bound to the number of preimages of iterates of $φ$ and $σ$. Using these new asymptotic upper bounds, a conjecuture in Konick and Katai's paper, "On the uniform distribution of certain sequences involving the Euler totient function and the sum of divisors function" is now proven and many corollaries follow from their proven conjecture.

The number of preimages of iterates of $φ$ and $σ$

Abstract

Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function and the sum-of-divisors functions . In this paper, we will extend the upper bound to the number of preimages of iterates of and . Using these new asymptotic upper bounds, a conjecuture in Konick and Katai's paper, "On the uniform distribution of certain sequences involving the Euler totient function and the sum of divisors function" is now proven and many corollaries follow from their proven conjecture.
Paper Structure (4 sections, 8 theorems, 59 equations)

This paper contains 4 sections, 8 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Bounds on some moment generating functions
  3. Proof of Theorem \ref{['New Pomerance upper bound']}
  4. Remarks

Key Result

Theorem A

As $n \rightarrow \infty$,

Theorems & Definitions (10)

  • Theorem A: Pomerance
  • Theorem 1
  • Theorem B
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Theorem 2