论文标题
在liouville功能上短时间
On the Liouville function in short intervals
论文作者
论文摘要
令$λ$表示Liouville功能。假设假设假设，我们证明了$ \ int_x^{2x} \ big | \ sum_ {x \ leq n \ leq n \ leq x+h}λ（n）\ big |^2 dx \ ll xh（\ log xh（\ log x）^6，$ x \ fefty $ xq \ exp \ left（\ sqrt {\ left（\ frac {1} {2} -o（1）\右）\ log x \ log x \ log x \ log x} \ right）。
Let $λ$ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $$\int_X^{2X}\Big|\sum_{x\leq n \leq x+h}λ(n) \Big|^2 dx \ll Xh(\log X)^6,$$ 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 Matom{ä}ki and Radziwiłł in their work on multiplicative functions in short intervals, as well as some standard results concerning smooth numbers.
