A+ CATEGORY SCIENTIFIC UNIT

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

Volume 123 / 2011

Youness Lamzouri, M. Tip Phaovibul, Alexandru Zaharescu 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 $3$.

Authors

  • Youness LamzouriDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    1409 W. Green Street
    Urbana, IL 61801, U.S.A.
    e-mail
  • M. Tip PhaovibulDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    1409 W. Green Street
    Urbana, IL 61801, U.S.A.
    e-mail
  • Alexandru ZaharescuDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    1409 W. Green Street
    Urbana, IL 61801, U.S.A.
    e-mail

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