2009 Number Theory Seminar - 06

The study of certain error terms arising in number theory can lead to very interesting results. For example, it was a great surprise when Littlewood discovered in 1914 that pi(x)-Li(x) changes sign infinitely often. Since then, finer questions have been asked about this error term, for example, what is the proportion of x such that pi(x)-Li(x) is positive. A similar phenomenon was observed by Chebyshev who noted that there seems to be more primes of the form 4n+3 than of the form 4n+1. Rubinstein and Sarnak gave a framework to study these questions in a groundbreaking article in 1994. We will push further their results, and show how one can compare different 'two-way prime number races' together, that is different error terms of the form pi(x;q,a)-pi(x;q,b), and see which is more often positive (or negative). The main tool is an asymptotic formula derived from the characteristic function of a random variable we will define. Here it is very interesting that these results are derived from a probabilistic model.