On the number of integers representable as sums of unit fractions. III. (Q1864866)

Let \(N(n)\) denote the set of all integers \(a\) that can be expressed as \(a= \sum_{1 \leq k \leq n} \frac{\varepsilon_k}{k}\), where \(\varepsilon_k \in \{0,1\}\). The author has previously shown [J. Number Theory 76, 206--216 (1999; Zbl 0926.11020), Erratum: ibid. 83, 183--184 (2000; Zbl 0926.11020)] that for some \(c >0\) \[ \log n+ \gamma -2 - \frac{c}{\log \log n} \leq | N(n)| \leq \log n + \gamma + O\left( \frac{ (\log \log n)^2}{\log n} \right). \] This was improved by \textit{E. S. Croot III} [Mathematika 46, 359--372 (1999; Zbl 1033.11014)] to \[ \log n+ \gamma - \left(\frac{9}{2}+o(1)\right) \frac{ (\log \log n)^2}{\log n} \leq | N(n)| \leq \log n + \gamma - \left( \frac{1}{2}+o(1)\right) \frac{ (\log \log n)^2}{\log n}. \] In the present paper the author proves that one can replace the \(\frac{9}{2}\) by \(\frac{\pi^2}{3}\).