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Statement

Definition (p. 603). "Let $S(N)$ denote the number of distinct values of $\sum_{k=1}^N\varepsilon_k/k$ where the $\varepsilon_k$'s take on all possible combinations of values with $\varepsilon_k=0$ or $1$." Here $\log_1x=\log x$ and $\log_jx=\log(\log_{j-1}x)$.

Lemma 5 (p. 603). For all $N\ge3$, $S(N)\ge2^{N/\log N}$.

Theorem 2 (p. 603). "If $r\ge1$ and $N$ is large enough that $\log_{2r}N\ge1$, then

$$S(N)\ \ge\ \exp\Bigl(\alpha\cdot\frac{N}{\log N}\cdot\prod_{j=3}^{r}\log_jN\Bigr)$$

where $\alpha=1/e$ is a permissible value for $\alpha$ and $\log_1x=\log x$, $\log_jx=\log(\log_{j-1}x)$." For $r\le2$ the product is empty.

Source. M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions II, Illinois J. Math. 20 (1976), 598--613; Section III, printed p. 603 (PDF p. 6), proof pp. 603--607. The introduction (p. 598) states the bound as $\frac{\alpha N}{\log N}\prod_{j=3}^r\log_jN\le\log S(N)$ for some $\alpha\ge1/e$. Read on the page image of p. 603; the scan's text layer garbles the displays.

Read depth. Claims checked: the definition, Lemma 5 and Theorem 2 were read clause by clause on the page image. The proof was read for its opening (below) and is not verified here.

Proof pointer

Lemma 5: distinct choices of the $\varepsilon_p$ over the primes $p\le N$ give distinct values, so $S(N)\ge2^{\pi(N)}$, and $\pi(N)\ge N/\log N$ for $N\ge17$ (Rosser--Schoenfeld); $3\le N\le16$ is checked directly. Theorem 2 is proved by induction on $r$ with the stronger inductive hypothesis (*) $S(N)\ge\exp\bigl(\prod_{j=3}^k(1-\frac3{\log_{2j-2}N})\cdot\frac{N}{\log N}\prod_{j=3}^k\log_jN\bigr)$ for $\log_{2k}N\ge1$, true for $k=1,2$ by Lemma 5; the step considers the primes $p$ with $Q=2N/\log N\le p\le N$ and the integers $k\le N$ with such a prime factor, and counts distinct values of $\sum_{p}\frac1p\sum_{k\le N/p}\varepsilon_k/k$ (pp. 603--607, read for structure only).

Dependencies

Rosser--Schoenfeld's explicit prime-counting bounds (the paper's [4]).

Bears on

• Problem 320: the 1976 lower bound for $\log S(N)$. It is weaker in the constant than the 1975 paper's Corollary 3 (constant $\log2$), which its own remark and Bettin, Grenié, Molteni and Sanna (who cite Theorems 2 and 3 together as [1, Th. 2 and 3], arXiv v1, p. 1) record; the 1975 paper was received in July 1974 and this one in July 1974 with a revision in January 1976, so the direction of improvement (the 1975 paper improves on this one, its reference [2]) follows the papers' own cross-references, not the publication dates.