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

• Autorzy: Youness Lamzouri , M. Tip Phaovibul , Alexandru Zaharescu
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11N69: Distribution of integers in special residue classes
• 11N64: Other results on the distribution of values or the characterization of arithmetic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2011
• Tom: 123
• Numer: 1
• Strony: 115-127

Daty

wydano

2011

Twórcy

autor

Youness Lamzouri

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

autor

M. Tip Phaovibul

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

autor

Alexandru Zaharescu

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

Identyfikatory

DOI

10.4064/cm123-1-8