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

BRÜDERN, JÖRG

Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 681-692

Résumé

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.

Détail

DOI : 10.1017/S0017089514000561

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