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Statement

Let $1=d_1(n)<d_2(n)<\cdots<d_{\tau(n)}(n)=n$ be the divisors of $n$. $D(x,t)$ is the number of positive integers $n\le x$ whose maximum ratio of consecutive divisors, $\max_{1\le i<\tau(n)}d_{i+1}(n)/d_i(n)$, is at most $t$ (p. 3); the integer $n=1$ is counted. The paper shows that $D(x,t)=B(x)$ for $\theta(n)=nt$ in the notation of Theorem 2. Put $v=\log x/\log t$. The function $d(v)$ is $0$ for $v<0$ and, for $v\ge0$, is given by

$$d(v)=1-\int_0^{\frac{v-1}{2}}\frac{d(u)}{u+1}\,\omega\Bigl(\frac{v-u}{u+1}\Bigr)\,du,$$

where $\omega$ is Buchstab's function (equation (4), p. 3, from the author's earlier paper); the paper also recalls $d(v)=\frac{C}{v+1}\{1+O((v+1)^{-2})\}$ for $v\ge0$, with $C=1/(1-e^{-\gamma})=2.280291\ldots$ (equation (5), p. 3).

Theorem 3 (p. 4). For $x\ge1$ and $t\ge2$,

$$D(x,t)=x\,\eta(t)\,d(v)\Bigl\{1+O\Bigl(\frac1{\log2x}\Bigr)\Bigr\},$$

where $0<\eta_0\le\eta(t)=1+O(1/\log t)$ for some positive constant $\eta_0$ (equation (6)).

Before the theorem the paper recalls Saias's bound $c_3x\log t/\log xt\le D(x,t)\le c_4x\log t/\log xt$ for $x\ge1$, $t\ge2$ (equation (2), p. 3, with $\log x$ replaced by $\log xt$ as its footnote says), and the author's earlier formula $D(x,t)=x\,d(v)\{1+O(1/\log t)\}$ for $x\ge t\ge\exp\{(\log\log x)^{5/3+\varepsilon}\}$ (equation (3)). Theorem 3 improves that error term and removes the lower bound on $t$, giving an asymptotic formula as $x\to\infty$ uniformly for $t\ge2$.

Corollary 3 (p. 4). For $x\ge t\ge2$, $D(x,t)=x\,d(v)\{1+O(1/\log t)\}$. The paper derives it from Theorem 3 and (6); it is (3) for every $t$ with $2\le t\le x$.

The paper's companion consequences are Corollary 1 (Theorem 3 with (5)) and Corollary 2, which follows from it: for $x\ge t\ge2$, $D(x,t)=\frac{Cx\log t}{\log xt}\{1+O(\frac1{\log t}+\frac{\log^2t}{\log^2x})\}$ (p. 4).

Source. Andreas Weingartner, Practical numbers and the distribution of divisors, Q. J. Math. 66 (2015), no. 2, 743--758, read in arXiv:1405.2585v3 (3 March 2015), as identified on the source card; the definitions on pp. 2--3, Theorem 3 and Corollaries 2 and 3 on p. 4, proved in Section 3 (pp. 6--10). The labels are the preprint's; the published version was not compared.

Read depth. Claims checked: the definitions, equations (2)--(6), Theorem 3 and Corollaries 2 and 3 were read clause by clause on the page images of pp. 3--4. The proof was read for its structure only and was not checked step by step.

Proof pointer

With $\theta(n)=nt$ the functional equation of Lemma 3 (p. 6) becomes Lemma 4: $D(x,t)=D(\sqrt{x/t},t)+[x]-\sum_{n\le\sqrt{x/t}}\chi_t(n)\Phi(x/n,nt)$ for $x\ge0$, $t\ge1$ (p. 6). The sieve estimate for $\Phi$ (Lemma 2) and Mertens' product, with Saias's bound (2) to control error terms, turn it into the integral equation of Lemma 8 (p. 8). In the variable $z$ with $x=t^{e^z-1}$ the normalized count $G_t(z)$ satisfies a convolution equation whose Laplace transform is compared with that of $G(z)=e^zd(e^z-1)$, which comes from (4); inverting gives equation (13), $D(x,t)=x\eta(t)d(v)+O(1+x\log t/(\log tx)^2)$, with $\eta(t)=\alpha(t)+\beta(t)$ (pp. 8--10). Since $d(v)\gg1/(v+1)$, the error is relative $O(1/\log2x)$, and the lower bound $\eta(t)\ge\eta_0$ for bounded $t$ comes from (13) and (2) (p. 10).