2011/07/27 by Chaohua Jia, Jia, Chaohua
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1107.5394
arxiv created 2011/07/27 · openalex publication_date 2011/07/27 · arxiv updated 2011/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the positive integer n, let f(n) denote the number of positive integer solutions (n1, n2, n3) of the Diophantine equation 4\over n=1\over n1+1\over n2+1\over n3. For the prime number p, f(p) can be split into f1(p)+f2(p), where fi(p)(i=1, 2) counts those solutions with exactly i of denominatorsn1, n2, n3 divisible by p. Recently Terence Tao proved that ∑p< xf2(p)≪ xlog2xloglog x with other results. But actually only the upper bound xlog2xloglog2x can be obtained in his discussion. In this note we shall use an elementary method to save a factor loglog x and recover the above estimate.