Daniel Fiorilli: Highly biased prime number races.

 In 1853, Chebyshev remarked that there are more primes of the form 4n+3 than of the form 4n+1 in the interval [1,x], for many values of x. Rubinstein and Sarnak established under GRH and a Linear Independence Hypothesis that the logarithmic density of x for which Chebyshev’s assertion is true is about 0.9959, and the logarithmic density of x such that Li(x)>pi(x) is about 0.99999973. Since their 1994 paper, many other densities have been computed and none of these numbers were found to exceed this last value. The goal of this talk will be to study a class of extremely biased prime number races, assuming a much weaker hypothesis than Linear Independence. If time permits, we will talk about races with elliptic curves, whose bias is determined by the relative size of the rank compared to the conductor.