The investigation of certain counting functions of elements with given factorization properties in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group. In this paper a constant arising from the investigation of the number of algebraic integers with factorizations of at most k different lengths is investigated. It is shown that this constant is positive if k is greater than 1 and that it is also positive if k equals 1 and the class group satisfies...

Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function$$N_f\left(x\right)=\#\left\{0\le n\<x\left|f\right(n)\mid n\right\}$$under a mild restriction on the values of $f$. When $f=s_q$, the base $q$ sum of digits function, the integers counted by $N_f$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.

We investigate the distribution of $\Phi \left(n\right)=1+{\sum }_{i=1}ⁿ\phi \left(i\right)$ (which counts the number of Farey fractions of order n) in residue classes. While numerical computations suggest that Φ(n) is equidistributed modulo q if q is odd, and is equidistributed modulo the odd residue classes modulo q when q is even, we prove that the set of integers n such that Φ(n) lies in these residue classes has a positive lower density when q = 3,4. We also provide a simple proof, based on the Selberg-Delange method, of a result of T. Dence and...

This article deals with the value distribution of multiplicative prime-independent arithmetic functions $\left(\alpha \right(n\left)\right)$ with $\alpha \left(n\right)=1$ if $n$ is $N$-free ($N\ge 2$ a fixed integer), $\alpha \left(n\right)>1$ else, and $\alpha \left(2^n\right)\to \infty$. An asymptotic result is established with an error term probably definitive on the basis of the present knowledge about the zeros of the zeta-function. Applications to the enumerative functions of Abelian groups and of semisimple rings of given finite order are discussed.

Let $J_k\left(n\right):=n^k{\prod }_{p\mid n}(1-p^{-k})$ (the $k$-th Jordan totient function, and for $k=1$ the Euler phi function), and consider the associated error term$$E_k\left(x\right):=\sum _{n\le x}{J}_k\left(n\right)-\frac{x^{k+1}}{(k+1)\zeta (k+1)}.$$When $k\ge 2$, both $i_k:=E_k\left(x\right)x^{-k}$ and $s_k:=lim\; sup{E}_k\left(x\right)x^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead$$Ik:=\underset{n\in \mathbb{N},n\to \infty }{lim\; inf}$$d 1 (d)dk ( 12 - { nd} ), since from [AS] $i_k=I_k-{\left(\zeta (k+1)\right)}^{-}1$ and from the present paper $s_k=-i_k$. We show that $I_k$ belongs to an interval of the form$$\left(\frac{1}{2\zeta \left(k\right)}-\frac{1}{(k-1)N^{k-1}},\frac{1}{2\zeta \left(k\right)}\right),$$where $N=N\left(k\right)\to \infty$ as $k\to \infty$. From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of $I_k$. We apply this algorithm...