Acta Mathematica Hungarica

, Volume 121, Issue 3, pp 293–305

Asymptotic behavior of the irrational factor


    • Department of MathematicsKoç University
  • A. H. Ledoan
    • Department of MathematicsUniversity of Rochester
  • A. Zaharescu
    • Department of MathematicsUniversity of Illinois at Urbana-Champaign

DOI: 10.1007/s10474-008-7212-9

Cite this article as:
Alkan, E., Ledoan, A.H. & Zaharescu, A. Acta Math Hung (2008) 121: 293. doi:10.1007/s10474-008-7212-9


We study the irrational factor function I(n) introduced by Atanassov and defined by \( I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } } \), where \( n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } \) is the prime factorization of n. We show that the sequence {G(n)/n}n≧1, where G(n) = Πν=1nI(ν)1/n, is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).

Key words and phrases

Irrational factorarithmetic functionsaveragesDirichlet seriesRiemann zeta-function

2000 Mathematics Subject Classification

primary 11M99secondary 11A2511A4111N37
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© Springer Science+Business Media B.V. 2008