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Random sums related to prime divisors of an integer. (English) Zbl 0722.11047

The authors investigate the asymptotic behavior of the sums \[ S(x)=\sum_{2\leq n\leq x}\frac{1}{\omega (n)}\sum_{p| n}f(p), \] where \(\omega\) (n) denotes the number of prime factors of n and the function f is of the form \(f(x)=x^{\rho}L(x)\) with a slowly oscillating function L(x). The nature of the estimates obtained depends on whether \(\rho <0\), \(\rho >0\), or \(\rho =0\). In the case \(\rho <0\), for example, the authors show that \[ S(x)=\sum^{M}_{j=0}\frac{c_ jx}{(\log \log x)^{j+1}}+O(\frac{x}{(\log \log x)^{M+2}}) \] holds with suitable constants \(c_ j\), for any given nonnegative integer M. This generalizes and sharpens an estimate of J.-M. De Koninck and J. Galambos [Acta Math. Hung. 52, 37-43 (1988; Zbl 0656.10043)] concerning the particular function \(f(x)=1/x\).

MSC:

11N37 Asymptotic results on arithmetic functions

Citations:

Zbl 0656.10043
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