Counting Egyptian fractions

Abstract

For any integer $N\ge 1$, let ${\mathfrak{E}}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of ${\mathfrak{E}}_N$, proving that $$\frac{N}{\log N}\prod _{j=3}^k{\log }_{j}N<\log (\#{\mathfrak{E}}_N)<0.421N,$$ for any fixed integer $k\ge 3$ and every sufficiently large $N$, where ${\log }_{j}x$ denotes the $j$-th iterated logarithm of $x$.

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