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Published online by Cambridge University Press: 15 May 2013
For any positive integer $n$, let
$f(n)$ denote the number of solutions to the Diophantine equation
$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$
$x, y, z$ positive integers. The Erdős–Straus conjecture asserts that
$f(n)\gt 0$ for every
$n\geq 2$. In this paper we obtain a number of upper and lower bounds for
$f(n)$ or
$f(p)$ for typical values of natural numbers
$n$ and primes
$p$. For instance, we establish that
$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$