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...