The Hurwitz and Riemann Ζ Functions

Manuel Eberl 🌐

October 12, 2017

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


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.


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Session Zeta_Function