Lethbridge Number Theory and Combinatorics Seminar: Farzad Aryan

The Riemann Hypothesis predicts that all zeros of the Riemann zeta function are located on the line Re(s) = 1/2. Also, we have that the number of zeros with imaginary parts located between T and 2T is approximately (T log T)/(2pi).
Therefore the average gap is about 2 pi/(log T). It has been conjectured that there are gaps that are smaller than 2 pi c/(\log T), for every c that is greater than 0. This has been proven for c slightly larger than 1/2.

Proving that c can be taken less than 1/2 seems to be a very hard problem, despite being far from the conjecture. In this talk we discuss the connection between Chowla's conjecture on the shifted convolution sums of the Liouville's function and the size of c.