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Statement

Setting. $B(x)$ and $D(x)$ are as in Theorem I and Theorem II, and $c_1$ is an absolute positive constant.

Theorem III (p. 258). For all sufficiently large values of $x$,

$$D(x)-B(x)>c_1\log\log\log x.$$

The paper also notes (p. 258 and its footnote) that infinitely many B-numbers are not D-numbers and infinitely many D-numbers are not B-numbers.

Source. P. Erdős and L. Mirsky, The distribution of values of the divisor function $d(n)$, Proc. London Math. Soc. (3) 2 (1952), 257--271; Theorem III on p. 258, its proof in §7, pp. 264--265. The copy read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images and the proof was read; its estimates were not re-derived. Nothing here is independently reviewed.

Proof pointer

§7, pp. 264--265. Each value of $d$ has one D-number $m$ and one B-number $m^*\ge m$ (p. 259), so $D(x)-B(x)$ counts the D-numbers $m\le x$ with $m^*>x$. Take $t$ with $p_1\cdots p_t\le x<p_1\cdots p_{t+1}$ and $r$ with $2^{2^r-1}<p_t<2^{2^{r+1}-1}$. For $3\le\nu\le r$ the integer $2^{2^\nu-1}p_2\cdots p_{t-1}$ is below $x$ and has $2^{t+\nu-2}$ divisors, so the D-number $m_\nu$ with that divisor count is at most $x$, while the B-number with that count is $p_1\cdots p_{t+\nu-2}>x$. This gives at least $r-2$ such D-numbers, and $r-2>c_1\log\log\log x$.

Dependencies

The correspondence between D-numbers and B-numbers set up on p. 259.

Bears on

No Erdős problem in the corpus; the theorem compares the two counts of Theorem IV.