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Estimates for certain sums involving the largest prime factor of an integer. (English) Zbl 0546.10037

Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 769-789 (1984).
[For the entire collection see Zbl 0541.00002.]
Let P(n) denote the largest prime divisor of n, and let \(S_ r(x)=\sum_{2\leq n\leq x}P^{-r}(n)\). The authors prove estimates for \(S_ r(x)\) and other related sums. A typical result is \[ S_ r(x)=x\{\exp (-2r)^{{1\over2}}\quad L(x)(1+g_{r-1}(x))+O(\log^ 3_ 3x/\log^ 3_ 2x)\}. \] Here, \(\log_ kx\) denotes the k-fold iterated logarithm, \(L(x)=(\log x \log_ 2x)^{{1\over2}}\), and \(g_ r(x)\) is a complicated (but explicitly given) function which satisfies \(g_ r(x)\sim\log_ 3x/2 \log_ 2x.\) The proof makes essential use of an improved estimate [due to E. R. Canfield, P. Erdős and the second author, J. Number Theory 17, 1-28 (1983; Zbl 0513.10043)] for \(\psi (x,y)=\sum_{n\leq x,P(n)\leq y}1.\)
Reviewer: S.W.Graham

MSC:

11N37 Asymptotic results on arithmetic functions
11N05 Distribution of primes