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.

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 |