Joint IAS/PU Number Theory

I will discuss recent work with Harald Helfgott in which we establish roughly speaking that the graph connecting nn to n±pn±p with pp a prime dividing nn is almost "locally Ramanujan". As a result we obtain improvements of results of Tao and Tao...

Sums of Dirichlet characters ?n?x?(n)?n?x?(n) (where ?? is a character modulo some prime rr, say) are one of the best studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya...