×

zbMATH — the first resource for mathematics

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)

MSC:
11N37 Asymptotic results on arithmetic functions
PDF BibTeX XML Cite