On the largest prime factor of an integer (Q690367)

If \(n\) has \(r\) prime factors, denote them by \(P_ r(n)\leq P_{r- 1}(n)\leq \cdots\leq P_ 2(n)\leq P_ 1(n)\), so that \(P_ k(n)\) (\(1\leq k\leq r\)) is the \(k\)-th largest prime factor of \(n\). The author first proves that, for any fixed integer \(A\geq 1\), there exist constants \(\lambda_ 2^{(j)}\) (\(j=1,\dots,A\)) such that \[ \sum_{n\leq x, \Omega(n)\geq 2} {1\over {P_ 2(n)}}=x \sum_{j=1}^ A {{\lambda_ 2^{(j)}}\over {\log^ j x}}+ O\biggl( {x\over {\log^{A+1}x}}\biggr), \tag{1} \] where \(\Omega(n)\) is the number of all prime factors of \(n\). More generally one has, for \(k\geq 3\) fixed, \[ \sum_{n\leq x, \Omega(n)\geq k} {1\over {P_ k(n)}}= \lambda_ k {{x(\log\log x)^{k- 2}}\over {\log x}} \left( 1+O \biggl( {1\over {\log\log x}}\biggr)\right) \tag{2} \] with \(\lambda_ k= \lambda_ 2^{(1)}/(k-2)!\). Formulas (1) and (2) sharpen the results of \textit{P. Erdős} and the reviewer [Publ. Inst. Math. 32, 49-56 (1982; Zbl 0506.10035)]. The author also evaluates the sum of reciprocals of \(P(n,Q)\), which denotes the largest prime factor of \(n\) belonging to \(Q\), where for some \(0<\delta<1\) and \(B>2\) \[ \sum_{p\leq x, p\in Q} 1=\delta \int_ 2^ x {{dt} \over {\log t}}+O \biggl( {x\over {\log^ B x}} \biggr). \] The related problems of the median value of \(P(n,Q)\) and \(P_ k(n)\) are also treated. The proofs, although in principle elementary, are technically rather complicated.