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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:
[1] Corput, J.G. van der: Une inégalité relative au nombre des diviseurs. Nederl. Akad. Wetensh. Proc. Ser. A42, 547–553 (1939) · JFM 65.0156.01
[2] Erdös, P.: On the sum \(\sum\limits_{k = 1}^x {d\left\{ {f\left( k \right)} \right\}} \) . J. Lond. Math. Soc.27, 7–15 (1952) · Zbl 0046.04103
[3] Erdös, P.: On the greatest prime factor of \(\prod\limits_{k = 1}^x {f\left( k \right)} \) . J. Lond. Math. Soc.27, 379–384 (1952) · Zbl 0046.04102
[4] Erdös, P., Schinzel, A.: On the greatest prime factor of \(\prod\limits_{k = 1}^x {f\left( k \right)} \) . Acta Arith. (à paraître)
[5] Hall, R.R., Tenenbaum, G.: The average orders of Hooley’s{\(\Delta\)} r -functions, II. Compos. Math.60, 163–186 (1986) · Zbl 0614.10037
[6] Hall, R.R., Tenenbaum, G.: Divisors. Cambridge: Cambridge University Press 1988
[7] Hooley, C.: On a new technique and its applications to the theory of numbers. Proc. Lond. Math. Soc.38, 115–151 (1979) · Zbl 0394.10027
[8] Landau, E.: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. (Leipzig: Teubner 1927); réimpression: New York: Chelsea 1949
[9] Landreau, B.: A new proof of a theorem of van der Corput. Bull. Lond. Math. Soc.21, 366–368 (1989) · Zbl 0677.10031
[10] Tenenbaum, G.: Fonctions {\(\Delta\)} de Hooley et applications. Séminaire de théorie des nombres, Paris 1984–85, Prog. Math.63, 225–239 (1986)
[11] Tenenbaum, G.: Sur une question d’Erdös et Schinzel. In: Baker, A., Bollobás, B., Hajnal, A. (eds.). A Tribute to Paul Erdös. Cambridge: Cambridge University Press (à paraître)
[12] Wolke, D.: Multiplicative Funktionen auf schnell wachsenden Folgen. J. Reine Angew. Math.251, 54–67 (1971) · Zbl 0234.10030
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