AI 中文总结
该研究针对短区间k自由整数上的指数和，证明了其s阶矩的紧界，由此得到带Möbius扭转的指数和的$L^1$均值下界，并指出改进相关$\ell^2$估计可进一步提升结果。
AI 中文摘要
设$S_k(\alpha;K)$表示短区间$(N-K,N]$中k自由整数上的指数和。对于$s>0$，当存在$\theta_{k,s}<1/2$使得$K \gg N^{\theta_{k,s}+\epsilon}$时，我们证明了$S_k(\alpha;K)$的s阶矩的本质上紧的界。作为直接结果，我们得到长度至少为$N^{0.49685}$的短区间上带Möbius函数扭转的指数和的$L^1$均值的下界。此外，我们表明，若改进涉及Möbius函数的$\ell^2$估计，所有这些结果都将立即得到进一步改进。
英文摘要
Let $S_k(α;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(α;K)$ whenever $K \gg N^{θ_{k,s}+ε}$ for some $θ_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the Möbius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the Möbius function.
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