On the mean value of the e-squarefree e-divisior function 1 by ProQuest

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									   Scientia Magna
  Vol. 5 (2009), No. 4, 36-40




On the mean value of the e-squarefree e-divisior
                 function 1
                                         Heng Liu and Yanru Dong


  School of Mathematical Sciences, Shandong Normal University, Jinan 250014, P.R.China
                  E-mail: sglheng@163.com dongyanru5252@163.com

    Abstract Let t(e) (n) denote the number of e-squarefree e-divisor of n. The aim of the present
    paper is to establish an asymptotic formula for the mean value of the function (t(e) )r , where
    r ≥ 1 is a fixed integer.
    Keywords E-squarefree e-divisor function, the generalized divisor function, convolution
    method.



§1. Introduction
                        s                                                     s
       An integer d = i=1 pbi is called an e-divisor of n = i=1 pai > 1 if bi |ai for every i ∈{1,
                               i                                       i
2, ..., s}, notation: d|e n. By convention 1|e 1. The integer n > 1 is called e-squarefree if all
exponents a1 , · · · as are squarefree. The integer 1 is also considered to be e-squarefree.
       Consider now the esponential squarefree exponential divisor (e-squarefre e-divisor)of n.
                s                                              s
Here d = i=1 pbi is an e-squarefree e-divisor of n = i=1 pai > 1, if b1 | a1 , · · · bs | as and
                     i                                              i
b1 , · · · bs are squarefree. Note that the integer 1 is e-squarefree but is not an e-divisor of n > 1.
       Let t(e) (n) denote the number of e-squarefree e-divior of n. The function t(e) is called the
                                                                         α1     αs
e-squarefre e-divisor function, which is multiplicative and if n = p1 · · · ps > 1, then (see [1])

                                         t(e) (n) = 2ω(α1 ) · · · 2ω(αs ) ,

where ω(α) = s denotes the number of distinct prime factors of α.
     a o o
    L´szl´ T´th [2] proved that the estimate
                                                                 1            1
                                        t(e) (n) = c1 x + c2 x 2 + O(x 4 + )                          (1)
                                  n≤x

holds for every   > 0, where
                                        ∞
                                            2ω(α) − 2ω(α−1)
                     c1 :=       (1 +                       ),
                             p          α=6
                                                  pα
                                               ∞
                             1      
								
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