The Hurwitz and Riemann ζ Functions

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.