On the Liouville Function in Short Intervals

Abstract

Abstract 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.