The Hurwitz and Riemann ζ Functions

This entry builds upon the results about formal and analytic Dirichlet series to define the Hurwitz ζ function ζ(a,s) and, based on that, the Riemann ζ function ζ(s). 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.

Apart from the most basic facts such as analyticity, the following results are provided:

• the Stieltjes constants and the Laurent expansion of ζ(s) at s = 1
• the non-vanishing of ζ(s) for ℜ(z) ≥ 1
• the relationship between ζ(a,s) and Γ
• the special values at negative integers and positive even integers
• Hurwitz's formula and the reflection formula for ζ(s)
• the Hadjicostas–Chapman formula

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.