Kathleen Petersen (FSU) The Riemann hypothesis is equivalent to a precise statement about the size of the error between the prime counting function and the logarithmic integral. Bombieri and Vinogradov's celebrated theorem shows that the average of this error term over arithmetic progressions is exactly what is predicted by the Riemann Hypothesis. I'll discuss a generalization of this to arbitrary number fields. This is joint work with M. Ram Murty.