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Statement

Notation (p. 121). Hooley's function is $\Delta(n)=\sup_u\operatorname{card}\{d: d\mid n,\ u<d\leq eu\}$, the largest number of divisors of $n$ in an interval $(u,eu]$. A relation holds p.p. when it holds on a sequence of integers of asymptotic density $1$.

Theorem 2 (p. 122). Let

$$\gamma<-\frac{\log2}{\log(1-1/\log3)}=0.28754\ldots.$$

Then $\Delta(n)>(\log\log n)^\gamma$ (p.p.).

For comparison the paper cites (p. 121) the upper bound $\Delta(n)\ll(\log n)^\beta$ p.p. for any $\beta>\log2\,(1-1/\log3)=0.06221\ldots$, from Hall and Tenenbaum.

Source. H. Maier and G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), no. 1, 121--128; Theorem 2 on p. 122, with the definition of $\Delta$ and the convention p.p. on p. 121. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the definitions were read clause by clause on the printed pp. 121--122. The proof (Section 4, pp. 126--128) was read for structure only, and nothing here is independently reviewed.

Proof pointer

Section 4, pp. 126--128. Fix $\rho>(1-1/\log3)^{-1}$, put $J=[(\log\log\log x)/\log\rho]$ and let $n_j$ be the product of the prime factors $p$ of $n$ with $\rho^j<\log\log p\leq\rho^{j+1}$, for $w(x)<j\leq J$. The argument of Theorem 1, run inside each block, shows that for every fixed $\eta>0$ almost all $n\leq x$ have, for every such $j$, divisors $d,d'$ of $n_j$ with $0<|\log(d'/d)|<\eta$. Choosing for each $j$ either $d$ or $d'$ gives $2^r$ distinct divisors of $n$, with $r=[J-w(x)]$, in an interval of logarithmic length $\eta r$, so the box principle gives $\Delta(n)\geq2^r/(\eta r)$. The exceptional set is $\ll xJw(x)^{-3}$, which is $o(x)$ for $w(x)=\sqrt{\log\log\log x}$.

Dependencies

Theorem 1 and its lemmas, adapted to the blocks $n_j$.

Bears on

• Problem 144: the theorem also implies the problem's statement, more weakly than Theorem 1. Fix $\gamma$ with $0<\gamma<0.28754\ldots$; then $(\log\log n)^\gamma\geq3$ for all large $n$, so for almost all $n$ some interval $(u,eu]$ holds three divisors $d_1<d_2<d_3$ of $n$. Since $d_3/d_1<e$, one of $d_2/d_1$ and $d_3/d_2$ is below $e^{1/2}<2$, so the set of $n$ with divisors $d<d'<2d$ contains a sequence of density $1$ and therefore has density $1$.