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Source. G. Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, in L. Lovász, I. Z. Ruzsa and V. T. Sós (eds), Erdős Centennial, Bolyai Society Mathematical Studies 25 (2013), 651--681. Labels and pages here are those of the author's version identified on the source card, paginated 1--22; the published chapter was not read. The statements are unnumbered: the lower bound on p. 8, the average estimates on p. 9.

Read depth. Claims checked: the statements were read clause by clause on the printed pages. The lower bound's two-line argument was read. The first average estimate is reported from Erdős and Tenenbaum 1983; the two-sided bound is obtained in the survey by combining Theorem 3 of that paper with (16), with no further detail.

Statement

Setting (p. 8). For $1=d_1<d_2<\cdots<d_{\tau(n)}=n$ the divisors of $n$,

$$G(n)=\sum_{1\le i<\tau(n)}\frac{d_i}{d_{i+1}}.$$

Erdős conjectured, in the passage the survey quotes, that $G(n)\to\infty$ outside a set of density $0$, and asked for an asymptotic formula for $\sum_{n\le x}G(n)$.

Lower bound (p. 8). If $p$ is the smallest prime factor of $n$, then $pd_i\mid n$ for at least $\frac12\tau(n)$ indices $i$, and so $G(n)>\tau(n)/2p$. In particular $G(n)>\tau(n)/\xi(n)$ for almost all $n$ whenever $\xi(n)\to\infty$, so $G(n)\to\infty$ for almost all $n$. The survey adds that this lower bound does not imply (9), the density-one statement for two divisors $d<d'<2d$.

Distribution (pp. 8--9). By Erdős and Tenenbaum (Bull. Soc. Math. France 111 (1983), 125--145), for every bounded real $\vartheta$ on $(0,1)$ the function $F(n;\vartheta)=\tau(n)^{-1}\sum_{1\le i<\tau(n)}\vartheta(d_i/d_{i+1})$ has a limiting distribution; in particular $G(n)/\tau(n)$ has one. It is not supported on $[\frac12,1]$: the survey states that $\mathrm d\{n\ge1:G(n)/\tau(n)\le\varepsilon\}>0$ for $0<\varepsilon\le1$, omitting the details.

Average estimates (p. 9). From the same paper, if $\vartheta$ is twice continuously differentiable on $[0,1]$,

$$\sum_{n\le x}F(n;\vartheta)=\vartheta(1)\,x\log x +O\Bigl(\frac{x(\log x)^{1-\delta}\log\log\log x}{\sqrt{\log\log x}}\Bigr),$$

with $\delta=1-(1+\log\log2)/\log2\approx0.08607$ as in (16) (p. 8), the exponent of $\log x$ being optimal. Combining Theorem 3 of that paper with Ford's estimate (16) for $H(x,y,2y)$ gives, for suitable positive constants $c_1,c_2$,

$$\frac{c_1x(\log x)^{1-\delta}}{(\log\log x)^{3/2}} \le x\log x-\sum_{n\le x}G(n) \le\frac{c_2x(\log x)^{1-\delta}}{(\log\log x)^{3/2}}.$$

No range of $x$ is printed; the bound is read for $x$ large.

Proof pointer

The lower bound is the one-line count above (p. 8). The distribution and average results are proved in Erdős and Tenenbaum 1983; (16) is from K. Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. (2) 168 (2008), 367--433 (card).

Dependencies

Erdős and Tenenbaum 1983 (Theorem 3 there) and Ford 2008, not proved in the survey.

Bears on

• Problem 673: the lower bound answers the first question yes, $G(n)\to\infty$ for almost all $n$, which the survey calls almost trivial; the two-sided bound gives $\sum_{n\le x}G(n)=x\log x+O\bigl(x(\log x)^{1-\delta}/(\log\log x)^{3/2}\bigr)$, an asymptotic formula with an error term of exact order, answering the second.