Joint IAS/Princeton University Number Theory Seminar

I will discuss recent work with Harald Helfgott in which we establish roughly speaking that the graph connecting $n$ to $n \pm p$ with $p$ a prime dividing $n$ is almost "locally Ramanujan". As a result we obtain improvements of results of Tao and Tao-Teravainen on logarithmic Chowla. I will discuss the main ideas in the proof and the connections with logarithmic Chowla.