Pacific Institute for the Mathematical Sciences - PIMS

The Chebyshev's Bias phenomenon from the point of view of probability theory and asymptotic formulas

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.

4:10pm-5:00pm, Room ASB 10900 (IRMACS)