On the distribution of the partial sum of Euler's totient function in residue classes
Volume 123 / 2011
Colloquium Mathematicum 123 (2011), 115-127
MSC: Primary 11N69; Secondary 11N64.
DOI: 10.4064/cm123-1-8
Abstract
We investigate the distribution of ${\mit\Phi} (n)=1+ \sum _{i=1}^n \varphi (i)$ (which counts the number of Farey fractions of order $n$) in residue classes. While numerical computations suggest that ${\mit\Phi} (n)$ 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 ${\mit\Phi} (n)$ 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 $\varphi (n)$ modulo $3$.
