# Prime number races

## Seminar/Forum

213
Peter Hall
This talk is a survey of "prime number races". Around 1850, Chebyshev noticed that for any given value of $$x$$, there always seem to be more primes of the form $$4n+3$$ less than $$x$$ than there are of the form $$4n+1$$. Similar observations have been made with primes of the form $$3n+2$$ and $$3n+1$$, primes of the form $$10n+3,10n+7$$ and $$10n+1,10n+9$$, and many others besides. More generally, one can consider primes of the form $$qn+a,qn+b,qn+c,\dots$$ for our favorite constants $$q,a,b,c,\dots$$ and try to figure out which forms are "preferred" over the others---not to mention figuring out what, precisely, being "preferred" means. All of these "races'' are related to the function $$\pi(x)$$ that counts the number of primes up to $$x$$, which has both an asymptotic formula with a wonderful proof and an associated "race'' of its own; and the attempts to analyze these races are closely related to the Riemann hypothesis---the most famous open problem in mathematics.