### Session S09 - Number Theory in the Americas

Thursday, July 15, 12:00 ~ 12:30 UTC-3

## Expansion, divisibility and parity

### Harald Helfgott

#### U. Gottingen, Germay - This email address is being protected from spambots. You need JavaScript enabled to view it.

We will discuss a graph that encodes the divisibility properties of integers by primes. We show that this graph is shown to have a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier. For instance: for $\lambda$ the Liouville function (that is, the completely multiplicative function with $\lambda(p) = -1$ for every prime), $$ \frac{1}{\log x} \sum_{n\leq x} \frac{\lambda(n) \lambda(n+1)}{n} = O\left(\frac{1}{{\sqrt{\log \log x}}}\right), $$ which is stronger than a well-known result by Tao. We also manage to prove, for example, that $\lambda(n+1)$ averages to 0 at almost all scales when $n$ is restricted to have a specific number of prime divisors $\Omega(n)=k$, for any "popular" value of $k$ (that is, $k = \log \log N + O({\sqrt{\log \log N}})$ for $n\le N$).

Joint work with M. Radziwill (Caltech, USA).