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A rate estimate in Billingsley’s theorem for the size distribution of large prime factors. (English) Zbl 1004.11050
For \(m\geq 1\), let \(P_j(m)\) \((j=1,\dots, \omega(m))\) denote the distinct prime factors of \(m\) where \(P_1(m)> P_2(m)>\dots> P_{\omega(n)}(m)\). For \(\vec\alpha_k= (\alpha_1,\dots, \alpha_k)\), let \[ F_n(\vec\alpha_k)= \upsilon_n \{m: P_j(m)> n^{\alpha_j},\;1\leq j\leq k\}. \] The objectives of this paper are to establish an asymptotic expansion in non positive powers of \(\log n\) for \(F_n(\vec\alpha_k)\) that is valid uniformly for \(\vec\alpha_k\) satisfying certain conditions and also to obtain an asymptotic formula that is valid for \(k\geq 2\) uniformly for \(\vec\alpha_k\in ]0,1]^k\). The latter result has an explicit estimate of the error term, thus improving a theorem equivalent to one of P. Billingsley [Period. Math. Hung. 2, 283-289 (1972; Zbl 0242.10033)] where the error term is just \(o(1)\). The author notes that a consequence of his result and its proof is the formula \[ \sum_{\substack{ m\leq n\\ P_k(m)\leq y}} 1=n r_k(u) \Biggl(1+ O\biggl( \frac{1}{\log y}\biggr) \Biggr), \qquad u= \frac{\log n}{\log y}, \] valid for fixed \(k\geq 2\) and uniformly for \(2\leq y\leq n\), where the function \(r_k(u)\) satisfies \(r_k(u) \asymp_k \frac{(\log 2u)^{k-2}}{u}\). An essential ingredient in the proof of the theorem is the asymptotic formula, in terms of the Dickman function \(\rho\), for \(\Psi(x,y)= \sum_{\substack{ m\leq x\\ P_1(m)\leq y}} 1\) established by E. Saias [J. Number Theory 32, 78-99 (1989; Zbl 0676.10028)] for the standard range \((\log\log x)^{\frac 53+ \varepsilon}\leq\log y\leq\log x\), \(x\geq 2\). The main term for \(F_n(\vec\alpha_k)\) can also be expressed as an integral involving \(\rho\).

MSC:
11N25 Distribution of integers with specified multiplicative constraints
11K65 Arithmetic functions in probabilistic number theory
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