Combinatorial Aspects of the Generalized Euler's Totient

A generalized Euler's totient is defined as a Dirichlet convolution of a power function and a product of the Souriau-Hsu-Möbius function with a completely multiplicative function. Two combinatorial aspects of the generalized Euler's totient, namely, its connections to other totients and its relations with counting formulae, are investigated

On the image of Euler's totient function

In this article we study certain properties of the image of Euler's totient function; we also consider the structure of the preimage of certain elements of the image of this function

Some remarks on Euler's totient function

The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not. If an even number is in the image, then its preimage can have at most half its elements odd. However, it may contain only even numbers. We consider the structure of the preimage of certain numbers in the image of the totient function.Comment: This article is a revision and extension of my previous article 'On the image of Euler's totient function' arXiv:0910.222

Euler's Totient Function

U ovom radu obradit ćemo Eulerovu funkciju i ukratko nešto reći o L. Euleru, matematičaru po kojemu je ta funkcija dobila ime. Definirat ćemo što je ta funkcija, navesti osnovna svojstva i neke ocjene te na primjerima pokazati kako se računa vrijednost te funkcije. Nadalje, navest ćemo primjene Eulerove funkcije i probleme usko vezane uz nju. Dokazat ćemo Eulerov teorem te također navesti njegove primjene.In this article we will cover the Euler’s totient function and briefly focus on L. Euler, a mathematician after whom this function is named. We will define this function, specify the basic properties and some bounds, and give some examples that show how to calculate the value of this function. Furthermore, we will list the applications and problems closely related to. We will prove Euler’s theorem and also indicate its application

Divisors of the Euler and Carmichael functions

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.Comment: v.3, 11 pages. To appear in Acta Arithmetica. Very small corrections and changes suggested by the referee. Added abstract, keywords, MS