Arithmetic functions in short intervals

par Yu-Chen Sun (University of Bristol, Royaume-Uni)

lundi 24 mars 2025, 14:00 → 15:00 Europe/Paris

Salle Pierre Grisvard (IHP - Bâtiment Borel)

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.