Article

Acta Mathematica Hungarica

, Volume 121, Issue 3, pp 293-305

Asymptotic behavior of the irrational factor

  • E. AlkanAffiliated withDepartment of Mathematics, Koç University Email author 
  • , A. H. LedoanAffiliated withDepartment of Mathematics, University of Rochester
  • , A. ZaharescuAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign

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Abstract

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) = Π ν=1 n I(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 factor arithmetic functions averages Dirichlet series Riemann zeta-function

2000 Mathematics Subject Classification

primary 11M99 secondary 11A25 11A41 11N37