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Statement

Setting (p. 257). Throughout the paper $p,q$ denote primes, $p_\nu$ is the $\nu$th prime, and $c_1,c_2,\ldots$ are absolute positive constants. An A-number is an integer $p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ with $a_1\ge a_2\ge\cdots\ge a_k$ and $k$ arbitrary; a B-number is an integer $p_1^{q_1-1}p_2^{q_2-1}\cdots p_k^{q_k-1}$ with primes $q_1\ge q_2\ge\cdots\ge q_k$ and $k$ arbitrary. $A(x)$ and $B(x)$ count the A-numbers and the B-numbers not exceeding $x$. The paper recalls Hardy and Ramanujan's asymptotic (1.1),

$$\log A(x)\sim\frac{2\pi}{\sqrt3}\Bigl(\frac{\log x}{\log\log x}\Bigr)^{1/2} \qquad(x\to\infty),$$

and the one-to-one correspondence $p_1^{a_1}\cdots p_k^{a_k}\leftrightarrow p_1^{p_{a_1}-1}\cdots p_k^{p_{a_k}-1}$ between A-numbers and B-numbers.

Theorem I (p. 257). As $x\to\infty$,

$$\log B(x)\sim\frac{2\pi\sqrt2}{\sqrt3}\,\frac{(\log x)^{1/2}}{\log\log x}.$$

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 I on p. 257, its proof in §§4--5, pp. 260--263. The copy read is identified on the source card.

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

Proof pointer

§§4--5, pp. 260--263. Lower bound (4.2): with $y=x^{(2-\epsilon)/\log\log x}$, the A-numbers up to $y$ are grouped by their part with large primes, which loses only a factor $\exp\{O((\log x)^{1/2}/(\log\log x)^2)\}$; a representative of each class with bounded exponents is sent to the B-number obtained by replacing each exponent $a$ by $p_a-1$, and the prime number theorem keeps that B-number below $x$; (1.1) then gives the bound. Upper bound (5.2): B-numbers up to $x$ are grouped by their part with large exponents, Lemma 1 (p. 260) bounds the size of each class, and the one representative of each class with all exponents large is sent to the A-number with exponents $\pi(a+1)$, which lies below $x^{(2+3\epsilon)/\log\log x}$; (1.1) again gives the bound.

Dependencies

Hardy and Ramanujan's asymptotic (1.1) for $A(x)$ (Proc. London Math. Soc. (2) 16 (1917), 112--132, the paper's cited source); the prime number theorem; Lemma 1 of the same paper (p. 260), which bounds by $(2n)^{2t}$ the number of non-increasing $n$-tuples of integers in $[0,t]$.

Bears on

The theorem is the input to Theorem II, which the paper uses for its upper bound on runs of distinct divisor counts in Problem 945; the theorem itself does not concern that problem's runs.