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}) any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid $\theta>7/12$. Ramachandra's corresponded to Huxley's $7/12$ exponent prime number theorem in short intervals. The main new idea leading improvement using Ramar\'e's identity extract small factor $n$-sum. proof method also allows us improve an estimate Zhan exponential sum as well some results multiplicative functions and almost primes