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 
$$\sum _{x<n\le x+h}\mu (n)=o(h)$$
holds for almost all $x\in [X,2X]$, where $h\to \mathrm{\infty }$ as $X\to \mathrm{\infty }$.
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 $x^{17/30+ϵ}<h\le x$, improved on Huxley's results from more than 50 years ago.