A remark on the Möbius function

Abstract

It is proved that for every positive $B$ there exist real numbers $0=a_0<a_1<\ldots <a_N=1$ and $ \max_{1\leq j\leq N} (a_{j-1}/a_j)\leq\theta<1$ such that $$\limsup_{x\to\infty} \frac{1}{\sqrt{x}}\sum_{j=1}^N\sum_{\theta a_j x<n\leq a_j x} \mu(n) \geq B$$ and $$\liminf_{x\to\infty}\frac{1}{\sqrt{x}}\sum_{j=1}^N\sum_{\theta a_j x<n\leq a_j x} \mu(n)\leq -B,$$ where $\mu(n)$ denotes the Möbius function.