Counting the Number of Solutions to the Erdős-straus Equation on Unit Fractions

For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation 4 n = 1 x + 1 y + 1 z with x, y, z positive integers. The Erdős-Straus conjecture asserts that f(n) > 0 for every n > 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 ∑ p<N f(p) asymptotically as N → ∞, where p ranges over primes. Our main result establishes the asymptotic upper and lower bounds N logN ∑ p6N f(p) N logN log logN. In particular, f(p) = Oδ(log 3 p log log p) for a subset of primes of density δ arbitrarily close to 1. Also, for a subset of the primes with density 1 the following lower bound holds: f(p) (log p)0.549. These upper and lower bounds show that a typical prime has a small number of solutions to the ErdősStraus Diophantine equation; small, when compared with other additive problems, like Waring’s problem. We establish several more results on f and related quantities, for instance the bound f(p) p 3 5 +O( 1 log log p ) for all primes p. Eventually we prove lower bounds for the number fm,k(n) of solutions of m n = 1 t1 + · · ·+ 1 tk , ∑ n6N fm,k(n) m,k N(logN) k−1−1 and a related result for primes.