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We investigate the distribution of (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 with if is -free ( a fixed integer), else, and . 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 (the -th Jordan totient function, and for the Euler phi function), and consider the associated error termWhen , both and are finite, and we are interested in estimating these quantities. We may consider insteadd 1 (d)dk ( 12 - { nd} ), since from [AS] and from the present paper . We show that belongs to an interval of the formwhere as . From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of . We apply this algorithm...
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