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Comparison of two dissimilar sums involving the largest prime factor of an integer. (English) Zbl 0853.11081
Berndt, Bruce C. (ed.) et al., Analytic number theory. Vol. 2. Proceedings of a conference in honor of Heini Halberstam, May 16-20, 1995, Urbana, IL, USA. Boston, MA: Birkhäuser. Prog. Math. 139, 723-735 (1996).
The author continues the study of the comparison of certain arithmetic sums, begun earlier [Acta Arith. 59, 339-363 (1991; Zbl 0731.11051)]. In the present work the sums \[ \Sigma (x)= \sum_{nf (P(n))\leq x} 1 \qquad \text{and} \qquad \Sigma' (x)= \sum_{n\leq x} {1\over {f(P(n))}} \] are compared, where \(P(n)\) is the largest prime factor of \(n\). The function \(f(t)\) (\(>0\) for \(t\geq 1\)) is continuous, strictly increasing (to \(+\infty\)) and satisfies (\(\nu \geq 0\) is fixed) \(\log f(t)= (\log t)^{\nu+ o(1)}\) \((t\to \infty)\).
The sums \(\Sigma (x)\) and \(\Sigma' (x)\) can be expressed in terms of the function \(\Psi (x, y)= \sum_{n\leq x, P(n) \leq y} 1\), whose asymptotic behaviour is well-known. Two theorems which compare the asymptotic behaviour of \(\Sigma (x)\) and \(\Sigma' (x)\) are proved, and they yield the following interesting corollary. If \(h(t):= (\log t)^{-1} \log f(t) \log \log t\) is monotonic, then \(\Sigma (x)\sim \Sigma' (x)\) as \(x\to \infty\) if and only if \(h(t)\to 0\) as \(t\to \infty\).
For the entire collection see [Zbl 0841.00015].
Reviewer: A.Ivić (Beograd)

11N37 Asymptotic results on arithmetic functions