Problem 320

Let $S(N)$ count the number of distinct sums of the form $\sum_{n\in A}\frac{1}{n}$ for $A\subseteq \{1,\ldots,N\}$ . Estimate $S(N)$ .

Source: erdosproblems.com/320

Number theory
Unit fractions

Wiki page

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SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved, in the site's label, which the site glosses as a resolution other than a proof or disproof, for the order of magnitude

\log S(N)\ \asymp\ \frac{N}{\log N}\prod_{j=3}^{k}\log_jN,\qquad \log_kN=O(1).

The lower bound of this order is refereed: Bettin, Grenié, Molteni and Sanna's Theorem 1 (Math. Comp., online 22 January 2026), on its claim page (Bettin Grenie Molteni Sanna, 2025). The earlier lower bounds of Bleicher and Erdős (Math. Comp. 1975; Illinois J. Math. 1976) hold only for $\log_kN\ge k$ and $\log_{2r}N\ge1$ respectively, and fall short of this order by an unbounded factor; the 1975 bound is on its claim page (Bleicher Erdos, 1975). The refereed upper bound (Bleicher and Erdős 1976, Theorem 3, on its claim page (Bleicher Erdos, 1976)) is weaker by the factor $\log_rN$ . The upper bound of the same order is a proof obtained with the AI system GPT 5.6 Sol Pro, submitted to the site's proof-claim tab on 15 July 2026 by Young, Zhu and Luo and accepted by the site as correct, with the site maintainer's exposition of 1 September 2026; its manuscript sits behind an Overleaf read link that served no document to a request on 2026-09-18, its Lean file covers finite combinatorial steps only, and no refereed publication or independent review of it was found on 2026-09-18. A second claim (Kominers and Neu, 22 July 2026) asserts a full asymptotic with a non-constant phase and has not been accepted. The two forum claims have their pages, the accepted order of magnitude and the pending asymptotic; with the three refereed bounds linked above, the standing derives from the five pages.

BlEr75: Bleicher, M. N. and Erdős, P., The number of distinct subsums of $\sum_{i=1}^N 1/i$ . Math. Comp. 29 (1975), no. 129, 29--42, DOI 10.1090/S0025-5718-1975-0366795-4; received 26 July 1974. Corollary 3, p. 40. Library home: Bleicher and Erdős 1975.
BlEr76b: Bleicher, M. N. and Erdős, P., Denominators of Egyptian fractions. II. Illinois J. Math. 20 (1976), 598--613; received 5 July 1974, revised 27 January 1976. Theorems 2 and 3, pp. 603 and 610.
BGMS25: Bettin, S., Grenié, L., Molteni, G. and Sanna, C., A lower bound for the number of Egyptian fractions. arXiv:2509.10030v1 (12 September 2025); Math. Comp., DOI 10.1090/mcom/4190, published online 22 January 2026 (Crossref record; no volume or pages assigned in it yet). Theorem 1, p. 2 of the preprint. Library home: Bettin et al. 2025.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), printed p. 43. Library home: Erdős and Graham 1980.
OEIS: Layman, J. W., Sequence A072207, The On-Line Encyclopedia of Integer Sequences (2002; entry last modified 21 October 2025, server time): $S(N)$ for $N\le83$ (b-file by B. Dobbelaere), linking [BlEr75] and [BGMS25]; accessed.
YZL26: Young, R., Zhu, K. and Luo, Y., manuscript behind the Overleaf read link given on the site's proof-claim tab (claim submitted 15 July 2026), which served no document to a request on 2026-09-18. Lean repository Zarathustra23/erdos-320-harmonic-subset-sums at its commit of 11 July 2026, pinned on the claim page.
KoNe26: Kominers, S. D. and Neu, J., The asymptotic number of distinct reciprocal subset sums. Manuscript, 56 pp., PDF dated 22 July 2026 on the first author's web site; Lean repository joachimneu/distinct-reciprocal-subset-sums at the commit of 22 July 2026 that the manuscript cites, pinned on the claim page. Not accepted by the site; the pending full claim a full asymptotic for log S(N).

Problem 320

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