  For any positive integer $n$, let $f(n)$ denote the number of solutions to
the Diophantine equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} +
\frac{1}{z}$ with $x,y,z$ positive integers. The \emph{Erd\H{o}s-Straus
conjecture} asserts that $f(n) > 0$ for every $n \geq 2$. To solve this
conjecture, it suffices without loss of generality to consider the case when
$n$ is a prime $p$.
  In this paper we consider the question of bounding the sum $\sum_{p<N} f(p)$
asymptotically as $N \to \infty$, where $p$ ranges over primes. Our main result
establishes the asymptotic upper and lower bounds $$ N \log^2 N \ll \sum_{p
\leq N} f(p) \ll N \log^2 N \log \log N.$$ In particular, from this bound and
the prime number theorem we have $f(p) = O(\log^3 p \log \log p)$ for a subset
of primes of density arbitrarily close to 1; thus a typical prime has a
relatively small number of solutions to the Erd\H{o}s-Straus Diophantine
equation.
  We also establish some related results on $f$ and related quantities, for
instance establishing the bound $f(p) \ll p^{3/5} + O(\frac{1}{\log\log p})}$
for all primes $p$.
