On the Möbius function in all short intervals

Kaisa Matomäki, Joni Teräväinen

We show that, for the Möbius function $\mu (n)$, we have $$\sum _{x<n\le x+x^{\theta }}\mu (n)=o(x^{\theta })$$ for any $\theta >0.55$. This improves on a result of Ramachandra from 1976, which is valid f…