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 C. Pomerance on the distribution of φ(n) modulo 3.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-1-8,
author = {Youness Lamzouri and M. Tip Phaovibul and Alexandru Zaharescu},
title = {On the distribution of the partial sum of Euler's totient function in residue classes},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {115-127},
zbl = {1245.11104},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-1-8}
}
Youness Lamzouri; M. Tip Phaovibul; Alexandru Zaharescu. On the distribution of the partial sum of Euler's totient function in residue classes. Colloquium Mathematicae, Tome 122 (2011) pp. 115-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm123-1-8/