On the Liouville function in short intervals

TL;DR

Under the assumption of the Riemann Hypothesis, the paper establishes an upper bound on the mean square of the Liouville function summed over short intervals, extending understanding of its behavior in such ranges.

Contribution

The paper proves a new bound on the second moment of the Liouville function in short intervals assuming RH, using simplified methods inspired by Matomäki and Radziwiłł.

Findings

Bound on the integral of squared sums of λ(n) in short intervals.

Extension of techniques to handle the Liouville function under RH.

Application of smooth number results in the analysis.

Abstract

Let $\lambda$ denote the Liouville function. Assuming the Riemann Hypothesis, we prove that $$\int_X^{2X}\Big|\sum_{x\leq n \leq x+h}\lambda(n) \Big|^2 dx \ll Xh(\log X)^6,$$ 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{\"a}ki and Radziwi{\l}{\l} in their work on multiplicative functions in short intervals, as well as some standard results concerning smooth numbers.