The number of solutions of the Erdos-Straus Equation and sums of k unit fractions

We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most O(n3/5+) solutions of m/n = 1/a1 + 1/a2 + 1/a3. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m = 4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time O(n (n3/m2)1/5), for any > 0. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given m. N in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation m/p = 1/a1 + 1/a2 + 1/a3 isf,m exp((5 log 2/(12 lcm(m, f)) + of,m(1)) log p/log log p). Previously, the best known lower bound of this type was of order (log p)0.549.