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### Abstract

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.