## 2009 Number Theory Seminar - 06

- Date: 02/12/2009

Simon Fraser University

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)