ON THE DISTRIBUTION OF THE PARTIAL SUM OF EULER'S TOTIENT FUNCTION IN RESIDUE CLASSES

We investigate the distribution of Phi(n) = 1 + Sigma(n)(i=1) phi(i) (which counts the number of Farey fractions of order n) in residue classes. While numerical computations suggest that Phi(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 Phi(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 C. Pomerance on the distribution of phi(n) modulo 3.