ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS

Abstract

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation (1)$ \begin{equation*} r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty). \end{equation*} $ Here ${\mathfrak s}$(n) is the singular series associated with sums of a square and three fifth powers, see (13) below for a precise definition. The main purpose of this note is to confirm (1) in mean square.