Expansion, divisibility and parity (joint work with M. Radziwill)

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 $\lambda(n)$ 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 $n$ 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 $n\le N.$ We will discuss the (mainly combinatorial, partly analytic) ideas behind the proof.
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000