The average number of solutions of the Diophantine equation U2+V2=W3and related arithmetic functions

Abstract

For the number of integer solutions of the title equation, withW≤;x (x a large parameter), an asymptotics of the form Ax log x + Bx + O(x1/2 (log x)3 (loglog x)2) is established. This is achieved in a general setting which furnishes applications to some other natural arithmetic functions.