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Sur une question d’Erdős et Schinzel. II. (On a question of Erdős and Schinzel. II). (French) Zbl 0699.10063
Let P(n) denote the largest prime factor of n, and let F(X) be a polynomial of degree $$g>1$$, with integer coefficients, which is irreducible in $${\mathbb{Z}}[X]$$. In a recent paper [“On the greatest prime factor of $$\prod^{x}_{k=1}f(k)$$”, Acta Arith. 55, No.2, 191-200 (1990)], P. Erdős and A. Schinzel proved that $P\left( \prod_{n\leq x} F(n) \right) > x \exp \exp (c(\log \log x)^{1/3}) \text{ for }x > x_ 0(F), \tag{*}$ $$c$$ being an absolute constant, a result not yet strong enough to confirm an announcement made by Erdős in 1952. In the present paper, the author establishes Erdős’ original announcement in a quantitative form by increasing the lower bound in (*) to $$x \exp((\log x)^\alpha)$$, where $$0 < \alpha < 2-\log4$$. This is achieved by deriving a lower bound for $H_ F (x,y,2y) = \text{card}\{n\leq x :\;\exists d | F(n),\quad y<d\leq 2y\}$ that is valid for $$y\leq x/2$$ ($$x,y\to\infty$$), and thus improving on results obtained in the author’s earlier paper of the same title [A Tribute to Paul Erdős, 405-443 (1990), edited by A. Baker, B. Bollobás and A. Hajnal, Cambridge University Press, Cambridge].
A key step, which is of interest in its own right, is the establishment of an upper bound for the sum $$\sum_{n\leq x}\Delta (F(n))^ t$$ for $$t\geq 1$$, where $$\Delta$$ denotes Hooley’s $$\Delta$$-function, defined by $$\Delta (n)=\max_{u\in {\mathbb{R}}}\text{card}\{d : \;d | n,\quad e^ u<d\leq e^{u+1}\}$$.
Reviewer: E.J.Scourfield

##### MSC:
 11N05 Distribution of primes 11N37 Asymptotic results on arithmetic functions 11B83 Special sequences and polynomials 11K65 Arithmetic functions in probabilistic number theory
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##### References:
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