Let $\lambda $ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $\int _X^{2X}\Big |\sum _{x\leq n \leq x+h}\lambda (n) \Big |^2{\textrm{d}} x \ll Xh(\log X)^6,$ as $X\rightarrow \infty $, provided $h=h(X)\leq \exp \left (\sqrt{\left (\frac{1}{2}-o(1)\right )\log X \log \log X}\right ).$ The proof uses a simple variation of the methods developed by Matomäki and Radziwiłł in their work on multiplicative functions in short intervals, as well as some standard results concerning smooth numbers.

中文翻译：

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短区间内的刘维尔函数

令 $\lambda $ 表示刘维尔函数。假设黎曼假设，我们证明 $\int _X^{2X}\Big |\sum _{x\leq n \leq x+h}\lambda (n) \Big |^2{\textrm{d}} x \ll Xh(\log X)^6,$ as $X\rightarrow \infty $，假设 $h=h(X)\leq \exp \left (\sqrt{\left (\frac{1}{2 }-o(1)\right )\log X \log \log X}\right ).$ 证明使用了 Matomäki 和 Radziwiłł 在其在短间隔内乘法函数的工作中开发的方法的简单变体，以及一些关于平滑数字的标准结果。