ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS
Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 681-692
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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.
BRÜDERN, JÖRG. ON THE ASYMPTOTIC FORMULA IN WARING'S PROBLEM: ONE SQUARE AND THREE FIFTH POWERS. Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 681-692. doi: 10.1017/S0017089514000561
@article{10_1017_S0017089514000561,
author = {BR\"UDERN, J\"ORG},
title = {ON {THE} {ASYMPTOTIC} {FORMULA} {IN} {WARING'S} {PROBLEM:} {ONE} {SQUARE} {AND} {THREE} {FIFTH} {POWERS}},
journal = {Glasgow mathematical journal},
pages = {681--692},
year = {2015},
volume = {57},
number = {3},
doi = {10.1017/S0017089514000561},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000561/}
}
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