On the Möbius function in all short intervals

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq
x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ for any $\theta>0.55$. This improves on a
result of Ramachandra from 1976, which is valid for $\theta>7/12$.
Ramachandra's result corresponded to Huxley's $7/12$ exponent for the prime
number theorem in short intervals. The main new idea leading to the improvement
is using Ramar\'e's identity to extract a small prime factor from the $n$-sum.
The proof method also allows us to improve on an estimate of Zhan for the
exponential sum of the M\"obius function as well as some results on
multiplicative functions and almost primes in short intervals.