Maier 1984 set divisors integer

../

theorem_1: For any function xi(n) tending to infinity, the least logarithmic ratio log(d'/d) of two distinct divisors of n is at most (log n)^(1-log 3) times exp(xi(n) sqrt(log log n)) for almost all n.

theorem_2: For every gamma below -log 2 / log(1 - 1/log 3) = 0.28754..., Hooley's function Delta(n) exceeds (log log n)^gamma for almost all n.

---

H. Maier and G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), no. 1, 121--128 (Oblatum 20-IX-1983; dedicated to Pál Erdős on the occasion of his 70th birthday); DOI 10.1007/BF01388495.

The copy read for this card is the Göttingen digitization (GDZ) of the article: a terms-of-use wrapper page (PDF p. 1, the only page with a text layer) followed by the eight printed pages 121--128 as page images without a text layer (PDF p. $n$ is printed p. $119+n$). The identity was confirmed on the page image of p. 121 (head "Invent. math. 76, 121--128 (1984)", title and authors). Provenance: obtained in September 2026; the wrapper names the article's persistent address within volume 76 (Werk Id PPN356556735_0076), http://resolver.sub.uni-goettingen.de/purl?PID=PPN356556735_0076|LOG_0015; 744,751 bytes. Read status: claims checked for Theorems 1 and 2 (statements read on the page images of pp. 121--122); Lemmas 1 and 2 (p. 123) were read as statements; the proofs (pp. 123--128) were read for structure only. That copy prints the digitizing library's terms on its wrapper page (PDF p. 1): access to the digitized documents is granted "strictly for noncommercial educational, research and private purposes", some of the library's collections "are protected by copyright", and "Publication and/or broadcast in any form (including electronic) requires prior written permission" from the library; a use permission for the Springer article that grants no redistribution right, every other right reserved.

Contents

The notation "p.p." (presque partout) means: for a sequence of asymptotic density $1$.

• Introduction (p. 121): Erdős's 45-year-old conjecture [1] that almost all integers possess a pair of divisors $d<d'\le2d$; Hooley's function $\Delta(n)=\sup_u\operatorname{card}\{d: d\mid n,\ u<d\le eu\}$; the best known results $x\log\log x\ll\sum_{n\le x}\Delta(n)\ll x(\log x)^\alpha$ with $\alpha=0.21969$, and $\Delta(n)\ll(\log n)^\beta$ p.p. for any $\beta>\log2\,(1-1/\log3)=0.06221\ldots$.
• Theorem 1 (p. 122; proof in section 3, pp. 123--126): let $E(n)$ be the infimum of the numbers $\log(d'/d)$ with $d\mid n$, $d'\mid n$, $d<d'$. If $\xi(n)$ is any function tending to infinity, then $E(n)\le(\log n)^{1-\log3}\exp\{\xi(n)\sqrt{\log\log n}\}$ p.p. The paper calls this nearly best possible: by Erdős--Hall [2] the exponent $1-\log3$ cannot be improved, and $\xi(n)$ cannot tend to $-\infty$ as fast as $-c\sqrt{\log\log\log\log n}$. A heuristic (p. 122) explains the exponent through the number $U(n)=\prod_{p^\nu\|n}(2\nu+1)$ of distinct ratios $d'/d$, which lies between $3^{\omega(n)}$ and $3^{\Omega(n)}$, and the normal order $\log\log n$ of $\omega(n)$ and $\Omega(n)$. Historical remark: the first author's original indirect proof gave $E(n)\le(\log n)^{1-\log3+\epsilon}$ p.p. for every $\epsilon>0$ through a comparison theorem; the paper presents the second author's number-theoretical proof.
• Theorem 2 (p. 122; proof in section 4, pp. 126--128): for $\gamma<-\log2/\log(1-1/\log3)=0.28754\ldots$, $\Delta(n)>(\log\log n)^\gamma$ p.p.
• Lemma 1 (p. 123), by the paper's account a weaker form of a theorem of Halberstam and Richert [6] that generalizes a result of Hall: if $f$ is nonnegative and multiplicative with $f(p^v)\le\lambda_1\lambda_2^v$ for all primes $p$ and $v\ge1$, where $\lambda_1>0$ and $0<\lambda_2<2$, then for $x\ge1$ $\sum_{n\le x}f(n)\ll x\prod_{p\le x}(1-p^{-1})\sum_{v\ge0}f(p^v)p^{-v}$, the implied constant depending on $\lambda_1$ and $\lambda_2$. Lemma 2 (p. 123): for $2\le u\le v\le x$, the number of $n\le x$ whose $u$-smooth part is at least $v$ is $\ll x\exp(-c\log v/\log u)$ for an absolute constant $c>0$. Lemmas 3 to 5 and a Corollary (pp. 124--125) are steps of the proof of Theorem 1.
• References (p. 128): fourteen items, including Erdős 1948 [1], Erdős--Hall 1979 "The propinquity of divisors" [2], Erdős--Tenenbaum 1981 and 1983 [4], [5], Hall--Tenenbaum [9], [10], Hooley 1979 [11] and Tenenbaum [12]--[14] (1979--1982, with [14] to appear).

Compiled scope

Pages 121--123 and 128 were read on the page images; pp. 123--128, the proofs of Theorems 1 and 2, were read for structure only. Result pages: Theorem 1 and Theorem 2, claims checked. Nothing was verified and nothing here is independently reviewed.

Bears on. #144: Theorem 1 implies the problem's statement in a stronger form, since $(\log n)^{1-\log3}\exp\{\xi(n)\sqrt{\log\log n}\}\to0$ for slowly growing $\xi$, so almost all $n$ have divisors $d<d'<2d$, and indeed, for each fixed $\beta<\log3-1$, with $d'/d<1+(\log n)^{-\beta}$; the set of such $n$ contains a sequence of asymptotic density $1$ by the meaning of "p.p.", so it has density $1$. Theorem 2 also implies the problem's statement, more weakly: for fixed $0<\gamma<0.28754\ldots$ it gives $\Delta(n)\geq3$ p.p., and three divisors in an interval $(u,eu]$ include two with ratio below $e^{1/2}<2$.

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