2018/05/08 by Christian Elsholtz, Elsholtz, Christian, Stefan Planitzer +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1805.02945
openalex publication_date 2018/05/08 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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 Oε(n3/5+ε) 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 Oε(nε((n3)/(m2))1/5), for any ε>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 \bmod f there exist infinitely many primes p such that the number of solutions of the equation (m)/(p)=(1)/(a1)+(1)/(a2)+(1)/(a3) is ≫f,m exp(((5log 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.