2009 Number Theory Seminar - 06

  • Date: 02/12/2009
Lecturer(s):
Daniel Fiorilli (Université de Montréal)
Location: 

Simon Fraser University

Topic: 

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

Description: 

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.

Schedule: 

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