On the Liouville function in short intervals

Abstract

Let $\lambda$ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $${\int }_X^{2X}|\sum _{x\le n\le x+h}\lambda (n){|}^{2}dx\ll Xh(\log X{)}^6,$$ as $X\to \infty$, provided $h=h(X)\le \exp \left(\sqrt{\left(\frac{1}{2}-o(1)\right)\log X\log \log X}\right).$ 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.

arXiv comment12 pages. Final version, published in IMRN