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Asymptotic behavior of the irrational factor

Authors:	E Alkan A H Ledoan A Zaharescu

Institution:	1.Department of Mathematics,Ko? University,Sariyer, Istanbul,Turkey;2.Department of Mathematics,University of Rochester,Rochester,USA;3.Department of Mathematics,University of Illinois at Urbana-Champaign,Urbana,USA

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=1ⁿ I(ν)^1/n, is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n). Research of the third author is supported in part by NSF grant number DMS-0456615.

Keywords:	and phrases" target="_blank"> and phrases Irrational factor arithmetic functions averages Dirichlet series Riemann zeta-function
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