关于十七个几乎相等的四次幂的华林问题中的渐近公式
Short Quartic Exponential Sums in an Intermediate Range and Waring's Problem for Fourth Powers with Almost Equal Summands
浏览论文内容
中文总结 AI 辅助
研究十七个几乎相等的四次幂的华林问题，利用哈代 - 利特伍德圆法及短四次指数和精细估计，证明更短区间渐近公式，改进了该问题中\(H\)的先前已知下界。
中文摘要 AI 辅助
本文考虑了十七个几乎相等的四次幂的华林问题。我们研究了一个足够大的自然数\(N\)表示为位于关于一个公共中心的短区间内的整数的十七个四次幂之和的表示数。利用哈代 - 利特伍德圆法和在有理逼近的中间范围内对短四次指数和的精细估计，我们证明了一个比之前已知的更短区间的渐近公式。该结果改进了十七个几乎相等的四次幂的华林问题中\(H\)的先前已知下界。
英文摘要
This paper is devoted to short quartic Weyl sums in an intermediate range of rational approximations and to their application to Waring's problem with almost equal summands. We obtain a pointwise estimate for such sums and use it, together with the Hardy--Littlewood circle method, to establish an asymptotic formula for the number of representations of a sufficiently large natural number as a sum of seventeen fourth powers of integers lying in a short interval about a common centre. The asymptotic formula is valid for $N^{6/25+\varepsilon}\leq H\leq N^{1/4-\varepsilon}$ and improves the previously known admissible lower bound for $H$.
