2011/09/11 by Jing-Jing Huang, Jingjing Huang, R. C. Vaughan +3
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1109.2274
9 pages, submitted
arxiv created 2011/09/11 · openalex publication_date 2011/09/11 · arxiv updated 2011/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we continue with our investigation of the Diophantine equation \fracan=\frac1x+\frac1y and in particular its number of solutions R(n;a) for fixed a. We prove a couple of mean value theorems for the second moment (R(n;a))2 and from which we deduce log R(n;a) satisfies a certain Gaussian distribution with mean log 3loglog n and variance (log 3)2loglog n, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of R(n;a) resembles the divisor function d(n2) in various aspects.