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Claim. Let $S(N)$ be the number of distinct values of $\sum_{n\in A}1/n$ over $A\subseteq\{1,\ldots,N\}$ and let $\log_j$ be the $j$-fold iterated natural logarithm. For every $k\ge4$ with $\log_kN\ge3/2$,

$$\log S(N)\ \ge\ 2\log2\,\frac{N}{\log N}\Bigl(1-\frac{3/2}{\log_kN}\Bigr)\prod_{j=3}^{k}\log_jN .$$

This is the third case of Theorem 1 of the paper (arXiv v1, p. 2); its first two cases give the factors $1$ when $\log_2N\ge1$ and $\log_3N$ when $\log_3N\ge1$. Because the condition $\log_kN\ge3/2$ admits every depth $k$ at which the iterated logarithm is still above a fixed constant, the bound has the order $\frac{N}{\log N}\prod_{j=3}^k\log_jN$ with $\log_kN=O(1)$, the lower half of the order of magnitude of Problem 320; the paper says that it improves the order of growth of Bleicher and Erdős's bounds, not only their constant (p. 2).

Covers. The lower bound for $\log S(N)$ of the right order. Not covered: the matching upper bound and any asymptotic.

Depends on. No page of this wiki; the theorem is the paper's own.

Acceptance. Refereed: the paper appeared in Mathematics of Computation, DOI 10.1090/mcom/4190, published online 22 January 2026; the page is dated by the arXiv posting of 12 September 2025. Reviewed: the site's curator, Thomas Bloom, labels the problem SOLVED for the order of magnitude and credits this bound in the commentary, noting that it grows faster than the 1975 bound; it is the lower half of that order, and the matching upper half is Young, Zhu and Luo's accepted claim, which takes its lower bound from this theorem. The curator is independent of the authors. The proof is not verified by this corpus.