Sur une question d'Erdős et Schinzel, II

../

theorem_1: Gives a uniform divisor-interval lower bound for irreducible integer polynomials of positive degree.

theorem_2: Gives a subpower exponential improvement over x for every irreducible integer polynomial of degree greater than one.

theorem_3: Bounds the t-th moments of Hooley's Delta function at the values of an irreducible integer polynomial, the estimate from which Theorem 1 is deduced.

---

Gérald Tenenbaum, Sur une question d'Erdős et Schinzel, II, Inventiones Mathematicae 99 (1990), 215--224, DOI 10.1007/BF01234418.

Copy read. The copy read for this card is a ten-page author-hosted publisher scan of printed pp. 215--224; its retrieval date is not recorded. This paper is a separate publication from the tribute chapter Sur une question d'Erdős et Schinzel at pp. 405--443. The scan prints "© Springer-Verlag 1990" in its first-page header, every other right reserved.

Theorem 1 is restated here for every positive-degree irreducible $F\in\mathbb Z[X]$ and every $\eta>\log4-1$:

$$H_F(x,y,2y)>x(\log x)^{-\eta}$$

as $x,y\to\infty$ with $y\leq x/2$. Here $H_F$, defined on printed p. 215 (physical p. 1), counts the integers $n\leq x$ for which $F(n)$ has a divisor $d$ with $y<d\leq2y$. Positive degree is an editorial restriction in this restatement: printed p. 216 states irreducibility without explicitly stating a degree condition, but constant prime polynomials would make the display false.

Inserting that estimate into equation (1.3) gives Theorem 2: for every irreducible $F\in\mathbb Z[X]$ of degree greater than one and every $0<\alpha<2-\log4$,

$$P\!\left(\prod_{n\leq x}F(n)\right) >x\exp\{(\log x)^\alpha\}$$

for each fixed $\alpha$ and all sufficiently large $x$; the source writes the threshold as $x_0(F)$ without asserting uniformity in $\alpha$. This is a theorem about the same running product as Problem 976, but $\exp\{(\log x)^\alpha\}=x^{o(1)}$ for the permitted $\alpha<1$. It therefore does not give a universal $x^{1+c}$ or $x^d$ bound. The positive-power results for $X^2+1$ reported on printed p. 405 of Paper I concern a quadratic special case.

The proof engine is Theorem 3 on printed p. 217, an averaged moment estimate for Hooley's divisor-concentration function evaluated at $F(n)$. Section 2 on printed pp. 217--222 supplies its proof; section 3 on printed p. 223 derives Theorem 1 using that estimate and (2.4). Immediately before Theorem 2 on printed p. 216, the paper identifies the running-product consequence through equation (1.3) on printed p. 215. This is a map of the source's argument, without a reconstruction of its external inputs.

Source: https://tenenb.perso.math.cnrs.fr/PPP/Erdos-Schinzel2.pdf.

Bears on. #976: Theorem 2 bounds below the greatest prime factor of the running product $\prod_{n\leq x}F(n)$ by $x\exp\{(\log x)^\alpha\}$ for every irreducible $F$ of degree greater than one and every fixed $0<\alpha<2-\log4$, for $x>x_0(F)$. The extra factor is $x^{o(1)}$, so it gives neither a bound $x^{1+c}$ with fixed $c>0$ nor a bound of order $x^g$; Theorems 1 and 3 enter only as its inputs.

Results to transcribe.

• Theorem 1: uniform divisor-interval lower bound.
• Theorem 2: general running-product lower bound with exact range $0<\alpha<2-\log4$.
• Theorem 3: moments of Hooley's $\Delta$ function at the values of an irreducible polynomial, the input to Theorem 1.

Living verification. Needs review. Printed pp. 215--224 were read for publication identity, definitions, theorem hypotheses and parameter ranges, formulas, and the proof map through Theorem 3 and section 3. The quadratic comparison was checked against I, printed p. 405. This is source-statement and dependency-map coverage; neither a complete proof reconstruction nor independent certification of the paper or its external inputs is supplied.

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