On the distribution of the Partial Sum of Euler's totient function in residue classes

Youness Lamzouri, M. Tip Phaovibul, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate the distribution of φ(n) = 1 + Σ n i=1 φ{symbol}(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{symbol}(n) modulo 3.

Original languageEnglish (US)
Pages (from-to)115-127
Number of pages13
JournalColloquium Mathematicum
Volume123
Issue number1
DOIs
StatePublished - 2011

Keywords

  • Distribution in residue classes
  • Euler's totient function

ASJC Scopus subject areas

  • Mathematics(all)

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