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Statement

Write $\sum_{1\le n\le x}1/n=m+\delta$, where $m=m(x)$ is the integer part and $\delta=\delta(x)$ the fractional part, and let $D(x)=(\tfrac12+o(1))(\log\log x)^2/\log x$. On p. 2 the paper records what the Main Theorem gives and what it leaves open: "We have trivially that $N(x)\subseteq\{1,2,\ldots,m\}$ and our main theorem tells us that when $x$ is sufficiently large, $\{1,2,\ldots,m-1\}\subseteq N(x)$. Moreover, if $\delta>((\tfrac92+o(1))(\log\log x)^2/\log x$ [sic] then $m\in N(x)$ so that $N(x)=\{1,2,\ldots,m\}$, and if $\delta<(\tfrac12+o(1))(\log\log x)^2/\log x$ then $m\notin N(x)$ so that $N(x)=\{1,2,\ldots,m-1\}$." Then the conjecture: "We believe that the upper bound in the Main Theorem is the truth, which if true would say that for $x$ sufficiently large,

$$N(x)=\begin{cases}\{1,2,\ldots,m\},&\text{if }\delta>D(x)\\ \{1,2,\ldots,m-1\},&\text{if }\delta<D(x).\end{cases}$$

"

Source. E. S. Croot III, On some questions of Erdős and Graham about Egyptian fractions, Mathematika 46 (1999), no. 2, 359--372; the author's typescript (14 pp.), p. 2, read on the page image. The statement is unnumbered in the paper; this page is named for its typescript page. The journal text was not compared.

Read depth. Claims checked: the passage was read clause by clause on the page image. It is a conjecture; the paper offers no proof of it, and the two-case description of $N(x)$ that precedes it follows from the Main Theorem as the paper says.

Dependencies

The Main Theorem of the same paper for the unconditional part; the conjecture itself is unproved.

Bears on

• Problem 308: the residue the Main Theorem leaves, $\delta$ between $(\tfrac12+o(1))(\log\log x)^2/\log x$ and $(\tfrac92+o(1))(\log\log x)^2/\log x$, is exactly where the smallest non-representable integer is undetermined; the conjecture would settle it.
• Problem 309: the same two-case description gives the count of representable integers as $m$ or $m-1$ for large $x$.