Arithmetic functions in short intervals

In this talk, we will first briefly review results related to arithmetic functions in short intervals, then we will focus on methods of the celebrated Matomaki-Radziwill theorem which shows that 
$$
\sum_{x< n \leq x+h } \mu(n) = o(h)
$$
holds for almost all $x \in [X,2X]$, where $h \to \infty$ as $X \to \infty$.
If time permits, we will introduce the recent breakthrough on primes in short intervals given by Maynard and Guth, where they proved that
$$
\pi(x+h)-\pi(x) \sim \frac{h}{\log x}
$$
for sufficiently large $x$ and $x^{17/30+ \epsilon}< h \leq x$, improved on Huxley's results from more than 50 years ago.