Slim Exceptional Sets for Sums of Cubes
Slim Exceptional Sets for Sums of Cubes
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DOI：
10.4153/cjm-2002-014-4
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发表时间：
2002-04
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通讯作者：
T. Wooley
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作者：
T. Wooley
摘要 我们研究与涉及立方和的各种加性问题相关的异常集。通过开发一种方法，其中在 Hardy-Littlewood 方法中明确采用例外集的指数和，我们能够更好地利用多余的变量。通过说明，我们表明不能被 9 整除且不超过 $X$ 且无法表示为 7 个素数立方之和的奇数整数的数量为 $O\left( {{X}^{23/36+\varepsilon }} \right)$ 。对于八个质数立方的和，对应的异常整数的数量是 $O\left( {{X}^{11/36+\varepsilon }} \right)$ 。
Abstract We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding $X$ , that fail to have a representation as the sum of 7 cubes of prime numbers, is $O\left( {{X}^{23/36+\varepsilon }} \right)$ . For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O\left( {{X}^{11/36+\varepsilon }} \right)$ .