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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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