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Set-theoretical entropies of Euler's totient function and other number theoretical special functions

Fatemah Ayatollah Zadeh Shirazi, Reza Yaghmaeian

TL;DR

The following text continues the studies on Alexandroff topologies induced by Euler's totient function and Dedekind psi function and pays attention to some of the other number theoretical special functions.

Abstract

In the following text we show set--theoretical entropy of Euler's totient function and contravariant set--theoretical entropy of Dedekind psi function are zero. Also contravariant set--theoretical entropy of Euler's totient function and set--theoretical entropy of Dedekind psi function are $+\infty$. We pay attention to some of the other number theoretical special functions too. We continue our studies on Alexandroff topologies induced by Euler's totient function and Dedekind psi function.

Set-theoretical entropies of Euler's totient function and other number theoretical special functions

TL;DR

The following text continues the studies on Alexandroff topologies induced by Euler's totient function and Dedekind psi function and pays attention to some of the other number theoretical special functions.

Abstract

In the following text we show set--theoretical entropy of Euler's totient function and contravariant set--theoretical entropy of Dedekind psi function are zero. Also contravariant set--theoretical entropy of Euler's totient function and set--theoretical entropy of Dedekind psi function are . We pay attention to some of the other number theoretical special functions too. We continue our studies on Alexandroff topologies induced by Euler's totient function and Dedekind psi function.

Paper Structure

This paper contains 3 sections, 12 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Infinite orbit number and infinite anti--orbit number of $\varphi$
  3. Some notes on Euler's totient function and Alexandroff topologies on $\mathbb N$

Key Result

Lemma 2.1

For $f:{\mathbb N}\to{\mathbb N}$ with $f(n)\leq n$ we have ${\mathfrak o}(f)=0$.

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 16 more