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Sur une question d’Erdős et Schinzel. (On a question of Erdős and Schinzel). (French) Zbl 0713.11069
A tribute to Paul Erdős, 405-443 (1990).
[For the entire collection see Zbl 0706.00007.]
Let F(X) denote a polynomial with integer coefficients and $$H_ F(x,y,z)$$ the number of integers $$n\leq x$$ for which there exists d $$|$$ F(n) with $$y<d\leq z$$. In [Acta Arith. 55, 191-200 (1990; Zbl 0715.11050)], P. Erdős and A. Schinzel derived a lower bound for $$H_ F(x,x/2,x)$$ in the course of establishing a lower bound for the largest prime factor of $$\prod_{n\leq x}F(n)$$ when F(X) is irreducible over $${\mathbb{Z}}[X]$$. In the present penetrating and interesting paper, the author advances the study of $$H_ F(x,y,z)$$ considerably by showing that, for $$z=y(1+(\log y)^{-\beta})$$ with $$0\leq \beta \leq B$$, $$y_ 0<y\leq x^ c$$ for some $$c<1$$, $$H_ F(x,y,z)$$ can be approximated by an expression of the form $$x(\log y)^{-\delta (\beta,F)}$$ for any polynomial F(X) as above that is positive whenever $$X\in {\mathbb{N}}$$ but is not necessarily irreducible; the non-negative number $$\delta$$ ($$\beta$$,F) is described precisely and is optimal. This generalizes Theorem 21 of [R. R. Hall and the author, Divisors (Cambridge Tracts in Mathematics 90) (1988; Zbl 0653.10001)] where the case $$F(X)=X$$ is investigated. The method does not extend to the case $$2y=z=x$$, which is dealt with by a different approach in Part II of this paper [Invent. Math. 99, 215-224 (1990; Zbl 0699.10063)] where the author also obtains an improvement in the result of Erdős and Schinzel referred to above.
Reviewer: E.J.Scourfield

##### MSC:
 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values