The Hurwitz and Riemann ζ Functions

 

Title: The Hurwitz and Riemann ζ Functions
Author: Manuel Eberl
Submission date: 2017-10-12
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.

BibTeX:
@article{Zeta_Function-AFP,
  author  = {Manuel Eberl},
  title   = {The Hurwitz and Riemann ζ Functions},
  journal = {Archive of Formal Proofs},
  month   = oct,
  year    = 2017,
  note    = {\url{https://isa-afp.org/entries/Zeta_Function.html},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License
Depends on: Bernoulli, Dirichlet_Series, Euler_MacLaurin, Winding_Number_Eval
Used by: Dirichlet_L, Prime_Distribution_Elementary, Prime_Number_Theorem