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Statement

Setting. $B(x)$ and $D(x)$ are as in Theorem I and Theorem II.

Theorem IV (p. 258). As $x\to\infty$,

$$\frac{D(x)}{B(x)}=1+O\Bigl\{\frac{(\log\log x)^2}{(\log x)^{1/3}}\Bigr\}.$$

The exponent $1/3$ is set small in the scan of p. 258; it is read as $1/3$ because the same glyph appears in (8.3) on p. 266, the bound the proof reduces Theorem IV to, and in the step $p_i^3<2\log x$, hence $p_i<2(\log x)^{1/3}$, on the same page.

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 IV on p. 258, its proof in §§8--10, pp. 265--269. The copy read is identified on the source card.

Read depth. Claims checked: the statement was read on the page images, with the exponent fixed as described above, and the proof was read in outline; its estimates were not re-derived. Nothing here is independently reviewed.

Proof pointer

§§8--10, pp. 265--269. Write $D(x)=B(x)+D_1(x)$ (8.1), where $D_1(x)$ counts the D-numbers $m\le x$ whose B-number $m^*$ exceeds $x$; such an $m$ has an exponent $a_i$ with $a_i+1$ composite (a critical exponent, at a critical prime $p_i$). Those with a critical prime below $2(\log x)^{1/3}$ are counted directly with Lemmas 1 and 2 (p. 260) and are few compared with $B(x)$ by Theorem I. For the rest, every critical exponent equals $3$ (p. 266); grouping them by their part with exponents above $3$ (the kernel), each kernel carries $O(\log x)$ of them (9.4), so (9.13) bounds $D_3(x)$, while §10 shows each such kernel is the kernel of many B-numbers up to $x$ (10.1), which gives (8.3) and so Theorem IV with (6.5).

Dependencies

Theorem I, the inequality $B(x)\le D(x)$ (6.5) from the proof of Theorem II, and Lemmas 1 and 2 (p. 260) of the same paper.

Bears on

No Erdős problem in the corpus.