On the distribution of totients 2 mod. 4

TL;DR

This paper investigates the distribution of totients congruent to 2 mod 4, revealing their asymptotic behavior, density properties, and connections to prime distributions, including cousin primes.

Contribution

It provides new asymptotic results on the distribution and density of totients 2 mod 4, and explores their relation to prime pairs and multiplicity properties.

Findings

The asymptotic magnitude of totients 2 mod 4 with multiplicity two is half that of primes.

The relative asymptotic density of certain totients with higher prime powers is zero.

Distribution results relate totients 2 mod 4 to cousin primes.

Abstract

In this paper we study the distribution of totients $2$ mod. $4$. We prove that the asymptotic magnitude of such totients with multiplicity two is half of that of prime numbers. As a corollary we obtain that the relative asymptotic density of the number of those totients with multiplicity four over the number of totients with multiplicity two is zero. We also obtain that the set of totients which have, a bigger than one, power of a prime in their pre-images and multiplicity $k$ has relative asymptotic density over the number of all totients of multiplicity $k$ equals to zero. A result on the distribution of consecutive pairs of totients $2$ mod. $4$, which relates to cousin primes, is also provide.