View abstract

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. document.getElementById('cloak877f1741d585544591f0b05c7fd4e05d').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy877f1741d585544591f0b05c7fd4e05d = 'h&#97;r&#97;ld.h&#101;lfg&#111;tt' + '&#64;'; addy877f1741d585544591f0b05c7fd4e05d = addy877f1741d585544591f0b05c7fd4e05d + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text877f1741d585544591f0b05c7fd4e05d = 'h&#97;r&#97;ld.h&#101;lfg&#111;tt' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak877f1741d585544591f0b05c7fd4e05d').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy877f1741d585544591f0b05c7fd4e05d + '\'>'+addy_text877f1741d585544591f0b05c7fd4e05d+'<\/a>';

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).