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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].