On the distribution of the partial sum of Euler's totient function in residue classes

Youness Lamzouri ¹   ; M. Tip Phaovibul ¹   ; Alexandru Zaharescu ¹

¹ Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green Street Urbana, IL 61801, U.S.A.

Colloquium Mathematicum, Tome 123 (2011) no. 1, pp. 115-127

Résumé

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 $3$.

Détail

DOI : 10.4064/cm123-1-8

Keywords: investigate distribution mit phi sum varphi which counts number farey fractions order residue classes while numerical computations suggest mit phi equidistributed modulo odd equidistributed modulo odd residue classes modulo even prove set integers mit phi lies these residue classes has positive lower density provide simple proof based selberg delange method result dence pomerance distribution varphi modulo