This entry builds upon the results about formal and analytic
Dirichlet series to define the Hurwitz ζ function and, based
on that, the Riemann ζ function. This is done by first
defining them for ℜ(*z*) > 1 and then
successively extending the domain to the left using the
Euler–MacLaurin formula. Some basic
results about these functions are also shown, such as their
analyticity on ℂ∖{1}, that they
have a simple pole with residue 1 at 1, their relation to the
Γ function, and the special values at negative integers and
positive even integers – including the famous ζ(-1)
= -1/12 and ζ(2) = π²/6.
Lastly, the entry also contains Euler's analytic proof of the
infinitude of primes, based on the fact that
ζ(*s*) has a pole at *s* =
1. |