EXCEPTIONAL SETS IN WARING’S PROBLEM: TWO SQUARES, TWO CUBES AND TWO SIXTH POWERS
EXCEPTIONAL SETS IN WARING’S PROBLEM: TWO SQUARES, TWO CUBES AND TWO SIXTH POWERS
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DOI：
10.11650/tjm.19.2015.5628
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发表时间：
2015-09
影响因子：
0.4
通讯作者：
Xiaodong Lü;Q. Mu
中科院分区：
文献类型：
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作者：
Xiaodong Lü;Q. Mu
令 $R(n)$ 表示大正整数 $n$ 的表示形式的数量，即两个平方、两个立方和两个六次方之和。本文证明，$R(n)$ 的预期渐近公式最多在 $O(\left( \log X \right)^{2+\varepsilon})$ 个不超过 $X$ 的正整数时失败。这是 T. D. Wooley 结果的改进，需要 $O(\left( \log X \right)^{3+\varepsilon})$。
Let $R(n)$ denote the number of representations of a large positive integer $n$ as the sum of two squares, two cubes and two sixth powers. In this paper, it is proved that the anticipated asymptotic formula of $R(n)$ fails for at most $O(\left( \log X \right)^{2+\varepsilon})$ positive integers not exceeding $X$. This is an improvement of T. D. Wooley's result which requires $O(\left( \log X \right)^{3+\varepsilon})$.