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Statement

Let $\theta$ be a real-valued arithmetic function. The set $\mathcal B$ contains $n=1$ and every $n\ge2$ with prime factorization $n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, $p_1<\cdots<p_k$, such that

$$p_j\le\theta\Bigl(\prod_{1\le i\le j-1}p_i^{\alpha_i}\Bigr)\qquad(1\le j\le k),$$

the empty product being $1$ (condition (1), p. 2). $B(x)$ is the number of $n\le x$ in $\mathcal B$.

Theorem 2 (p. 2). Suppose $\theta(1)\ge2$ and

$$n\le\theta(n)\le An(\log2n)^a(\log\log3n)^b\qquad(n\ge1)$$

for constants $A,a,b$ with $A\ge1$ and $0\le a\le1$.

• If $a<1$, then $B(x)=\dfrac{c_\theta x}{\log x}\bigl\{1+O\bigl((\log x)^{a-1}(\log\log x)^b\bigr)\bigr\}$ for $x\ge3$.
• If $a=1$ and $b<-1$, then $B(x)=\dfrac{c_\theta x}{\log x}\bigl\{1+O\bigl((\log\log x)^{b+1}\bigr)\bigr\}$ for $x\ge3$.

In either case $c_\theta$ is a positive constant depending on $\theta$, and the implied constant in the error term depends on $A$, $a$ and $b$.

The paper names three cases (p. 2): $\theta(n)=\sigma(n)+1$ gives the practical numbers, $B(x)=P(x)$, with $(a,b)=(0,1)$ (Theorem 1); $\theta(n)=n+2$ gives Thompson's weakly $\varphi$-practical numbers, with $(a,b)=(0,0)$; and $\theta(n)=nt$ gives $B(x)=D(x,t)$, the count of $n\le x$ whose consecutive divisors have ratio at most $t$ (p. 3), so that for fixed $t$ Theorem 2 with $(a,b)=(0,0)$ gives an asymptotic estimate for $D(x,t)$.

Theorem 4 (pp. 11--12), the paper's more general form: if $\theta(1)\ge2$ and $n\le\theta(n)\le nf(n)$ for $n\ge1$, where $f$ is non-decreasing, $(\log f(x))^2/\log2x$ is decreasing for sufficiently large $x$, and $f(x)\ll\log2x/(\log\log3x)^{1+\varepsilon}$ for $x\ge1$ and some $\varepsilon>0$ (condition (18)), then with $h(x)=\int_x^\infty f(y)\,y^{-1}(\log2y)^{-2}\,dy$ there is a positive constant $c_\theta$ depending on $\theta$ such that $B(x)=(c_\theta x/\log x)\{1+O(h(x))\}$ for $x\ge2$. Theorem 2 is the case $f(x)=A(\log2x)^a(\log\log3x)^b$ (p. 12).

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; Theorem 2 on p. 2; Theorem 4 on pp. 11--12, proved in Section 5 (pp. 11--15). The labels are the preprint's; the published version was not compared.

Read depth. Claims checked: Theorems 2 and 4 and the three named cases were read clause by clause on the page images of pp. 2--3 and 11--12. The proof was read for its structure only and was not checked step by step.

Proof pointer

The tool is the functional equation of Lemma 3 (p. 6): if $\theta(n)\ge P^+(n)$, the largest prime factor, then $[x]=\sum_{n\le x}\chi(n)\Phi(x/n,\theta(n))$ for $x\ge0$, where $\chi$ is the indicator of $\mathcal B$ and $\Phi(x,y)$ counts the $n\le x$ with no prime factor up to $y$; it comes from writing each $m\le x$ uniquely as $m=nr$ with $n\in\mathcal B$ and every prime factor of $r$ above $\theta(n)$. Lemma 9 (p. 12) gives the first bounds $x/\log2x\ll B(x)\ll x\log f(x)/\log2x$ by comparison with $D(x,t)$ and Saias's estimate. The sieve estimate of Lemma 2 for $\Phi$ in terms of Buchstab's function $\omega$ and Mertens' product turns the functional equation into an integral equation for $B(x)$ (Lemmas 10--15, pp. 12--14); in the variable $z$ with $x=2^{e^z-1}$, a Laplace transform compares it with equation (4) (p. 3) for the function $d(v)$ of Theorem 3, and $\lceil1/\varepsilon\rceil$ rounds of the resulting estimate remove the provisional upper bound, giving Theorem 4 (p. 15).