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

Youness Lamzouri, M. Tip Phaovibul, and Alexandru Zaharescu. "On the distribution of the partial sum of Euler's totient function in residue classes." Colloquium Mathematicae 123.1 (2011): 115-127. <http://eudml.org/doc/284084>.

@article{YounessLamzouri2011,
abstract = {We investigate the distribution of $Φ(n) = 1+ ∑_\{i=1\}ⁿ φ(i)$ (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.},
author = {Youness Lamzouri, M. Tip Phaovibul, Alexandru Zaharescu},
journal = {Colloquium Mathematicae},
keywords = {Euler's totient function; distribution in residue classes},
language = {eng},
number = {1},
pages = {115-127},
title = {On the distribution of the partial sum of Euler's totient function in residue classes},
url = {http://eudml.org/doc/284084},
volume = {123},
year = {2011},
}

TY - JOUR
AU - Youness Lamzouri
AU - M. Tip Phaovibul
AU - Alexandru Zaharescu
TI - On the distribution of the partial sum of Euler's totient function in residue classes
JO - Colloquium Mathematicae
PY - 2011
VL - 123
IS - 1
SP - 115
EP - 127
AB - We investigate the distribution of $Φ(n) = 1+ ∑_{i=1}ⁿ φ(i)$ (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.
LA - eng
KW - Euler's totient function; distribution in residue classes
UR - http://eudml.org/doc/284084
ER -