, Volume 121, Issue 3, pp 293305
First online:
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 UrbanaChampaign
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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 zetafunction2000 Mathematics Subject Classification
primary 11M99 secondary 11A25 11A41 11N37 Title
 Asymptotic behavior of the irrational factor
 Journal

Acta Mathematica Hungarica
Volume 121, Issue 3 , pp 293305
 Cover Date
 200811
 DOI
 10.1007/s1047400872129
 Print ISSN
 02365294
 Online ISSN
 15882632
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Irrational factor
 arithmetic functions
 averages
 Dirichlet series
 Riemann zetafunction
 primary 11M99
 secondary 11A25
 11A41
 11N37
 Authors

 E. Alkan ^{(1)}
 A. H. Ledoan ^{(2)}
 A. Zaharescu ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Koç University, 34450, Sariyer, Istanbul, Turkey
 2. Department of Mathematics, University of Rochester, Rochester, New York, 14627, USA
 3. Department of Mathematics, University of Illinois at UrbanaChampaign, 273 Altgeld Hall, MC382, 1409 W. Green Street, Urbana, Illinois, 61801, USA