Sur une question d'Erdős et Schinzel

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irreducible_interval_corollary: Gives the short-interval divisor asymptotic for irreducible polynomials and characterizes when the associated density tends to zero.

main_theorem: Gives two regimes for the count of polynomial values having a divisor in a prescribed short interval.

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Gérald Tenenbaum, Sur une question d'Erdős et Schinzel, in A. Baker, B. Bollobás, and A. Hajnal (eds.), A Tribute to Paul Erdős, Cambridge University Press (1990), visibly printed pp. 405--443 in the reprint read, DOI 10.1017/CBO9780511983917.035.

Copy read. The copy read for this card is the author-corrected reprint, which has 39 physical pages and explicitly says that it includes corrections to the published version. Its visible article pages are 405--443; Crossref's 405--444 span is retained only as differing bibliographic metadata. The reprint, an author-typeset copy from the author's site that reproduces the publisher's line, prints "© Cambridge University Press, 1990." on its first page, every other right reserved.

For $x\geq z\geq y\geq2$, the paper writes $H_F(x,y,z)$ for the number of $n\leq x$ such that $F(n)$ has a divisor $d$ with $y<d\leq z$. Its main theorem gives two regimes of estimates for this count when $z=y\{1+(\log y)^{-\beta}\}$. Its Complement says that, after changing the values of $y_0$ and $c_j$ ($1\leq j\leq5$), the lower bound in (1.13) and both bounds in (1.14) hold throughout $y\leq x^{c_0}$ for every fixed $c_0<1$; it does not include the upper bound in (1.13) in that extension. The irreducible specialization and corollary give

$$H_F(x,y,2y)=x(\log y)^{-\delta+o(1)}, \qquad \delta=1-\frac{1+\log\log2}{\log2},$$

in that sublinear-$y$ range, and characterize when the corresponding density tends to zero.

Printed p. 405 (physical p. 1) also reports earlier positive-power progress for the special polynomial $X^2+1$: Hooley obtained exponent $1+1/10$, and Deshouillers--Iwaniec improved the added exponent to a value slightly larger than $1/5$. These are historical reports of results proved in the cited papers, not theorems proved in this chapter. They give positive-power progress for the quadratic special case of Problem 976.

The note added on printed p. 442 (physical p. 38) points to the distinct paper Sur une question d'Erdős et Schinzel, II. It records a divisor estimate for every irreducible $F\in\mathbb Z[X]$. The running-product consequence (1.17) is stated for irreducible polynomials of degree greater than one in Tenenbaum II, Theorem 2, printed p. 216 (physical p. 2), with every exponent $0<c<2-\log4$. The two publications and their page systems are not merged.

Paper I itself explains on printed p. 408 that its method cannot take $y$ of order $x$. Even the hypothetical extension to $y=x$ discussed there yields only (1.17), a lower-bound threshold $x\exp\{(\log x)^c\}$ with $c<1-\delta<1$. This is $x^{1+o(1)}$, so it does not establish E976's $x^{1+c'}$ bound for any fixed $c'>0$. Paper I is source and method context for E976, not a universal positive-power theorem.

Source: https://tenenb.perso.math.cnrs.fr/PPP/Erdos-Schinzel%2C1.pdf.

Bears on. Problem 976: source and method context. The paper's estimates count divisors of polynomial values in short intervals below $x^{c_0}$, $c_0<1$; it reports the Hooley and Deshouillers--Iwaniec positive-power results for $X^2+1$ from the cited papers, and its own results give no positive-power lower bound for the greatest prime factor of $\prod_{n\leq x}F(n)$.

Results to transcribe.

• Main theorem: the two-regime estimate for $H_F(x,y,z)$ on printed pp. 407--408.
• [[arithmetic_functions/tenenbaum_1990_sur_une_question_erdos_schinzel_i/irreducible_interval_corollary|Irreducible specialization and corollary]]: the short-divisor asymptotic and the criterion for $D_F(y)\to0$ on printed p. 408.

Living verification. Needs review. Printed pp. 405--408 and 442 were checked for the edition's identity, historical quadratic exponents, result statements, method limitation, and postscript. The cross-paper running-product restriction was checked in II, Theorem 2, printed p. 216. These are source-statement and locator checks; the cited quadratic proofs and the complete chapter proof were not reconstructed or independently certified.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.