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

TL;DR

This paper investigates the set-theoretical entropies of Euler's totient and Dedekind psi functions, revealing their entropy values and exploring related topologies, thus advancing understanding of their dynamical properties.

Contribution

It provides new results on the set-theoretical entropies of key number-theoretic functions and examines the induced topologies, contributing to the theoretical understanding of their dynamical behavior.

Findings

Entropy of Euler's totient function is zero and infinite in different contexts.

Entropy of Dedekind psi function is zero and infinite in different contexts.

The study extends to other number-theoretic special functions and their associated topologies.

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.