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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

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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