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 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.

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{http://isa-afp.org/entries/Zeta_Function.html},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License
Depends on: Bernoulli, Dirichlet_Series, Euler_MacLaurin