Some Properties of Euler’s Function and of the Function τ and Their Generalizations in Algebraic Number Fields

In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields.

Some Properties of Extended Euler’s Function and Extended Dedekind’s Function

In this paper, we find some properties of Euler’s function and Dedekind’s function. We also generalize these results, from an algebraic point of view, for extended Euler’s function and extended Dedekind’s function, in algebraic number fields. Additionally, some known inequalities involving Euler’s function and Dedekind’s function, we generalize them for extended Euler’s function and extended Dedekind’s function, working in a ring of integers of algebraic number fields.

The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$ , $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$ .

The shifted sum of the first n values of Euler’s function

In this paper, we prove that, for any given odd number [Formula: see text], the number of integers [Formula: see text] such that [Formula: see text] is a square is [Formula: see text], where [Formula: see text] is the Euler totient function. Beyond this, we pose a conjecture and a problem for further research.

On Euler's function

By giving some new treatments we can improve a classical result of Walfisz (1963) on the asymptotic formula of Euler's function.

Consecutive coincidences of Euler’s function

We prove a version of the Hardy–Ramanujan inequality and a bound on the count of smooth numbers up to some number [Formula: see text], both with explicit constants. We use these as tools to prove a few interesting results on the values [Formula: see text] satisfying [Formula: see text] and provide an explicit bound on the sum of the reciprocals of such [Formula: see text].