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Claim. For an infinite set $A$ of positive integers write $f_A(x)=\sum_{a\in A,\,a\le x}1/a$ and $D_A(x)=\max_{n\le x}d_A(n)$, where $d_A(n)$ counts the elements of $A$ dividing $n$. Part II of the Erdős–Sárközy series on generalized divisor functions proves that $f_A(x)\to\infty$ implies

$$\limsup_{x\to\infty}\frac{D_A(x)}{\exp\bigl(c_1(\log f_A(x))^2\bigr)}=\infty$$

for an absolute constant $c_1>0$, through the local statement that $f_A(x)>\exp((\log\log x)^{1/2})$ forces $D_A(y)>\exp(c_2(\log f_A(x))^2)$ at $y=\exp((\log x)^2)$. Since $\exp(c_1(\log f)^2)/f^k\to\infty$ as $f\to\infty$ for every fixed $k$, the ratio in [[problems/divisors/E0444/_index|Problem 444]] is unbounded for every $k$ when $f_A(x)\to\infty$; when $f_A(x)$ stays bounded, Part I of the series, which proves $\limsup D_A(x)/f_A(x)=\infty$ for every infinite $A$, already makes $D_A(x)$ unbounded against a bounded denominator. The answer is therefore yes for every $k$. The problem's numerator ranges over $n<x$ and its sum over $a<x$ where the papers use $n\le x$ and $a\le x$; the limits superior are unaffected.

The site credits Part IV of the series, Studia Sci. Math. Hungar. 15 (1980), 467–479 (card). Its introduction restates the displayed theorem as proved in Part II and records Part I's result; its own Theorem 2 concerns a different question, the smallest $y$ with $D_A(y)>\Omega f_A(x)$, and gives only a constant multiple of $f_A(x)$. The proof of the displayed theorem is in Part II, J. Number Theory 15 (1982), no. 1, 115–136, the second link. Erdős and Graham posed the question in their 1980 problem book, p. 88, recording the $k=1$ case as proved by Erdős and Sárközy and the general $k$ as something they believed but could not prove (card).

Acceptance. Refereed: Studia Sci. Math. Hungar. 15 (1980), 467–479, and J. Number Theory 15 (1982), no. 1, 115–136. Reviewed: the site's curator, T. F. Bloom, marks Problem 444 proved and credits Part IV. Part IV appears in the volume dated 1980 but was received on 25 September 1981 and cites Part II as J. Number Theory 15 (1982); the page is dated by Part II's issue, August 1982, where the theorem was first published. No formalization is recorded, and this repository has not checked the proof independently.