
The
Hurwitz
and
Riemann
ζ
Functions
Title: 
The Hurwitz and Riemann ζ Functions 
Author:

Manuel Eberl

Submission date: 
20171012 
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_FunctionAFP,
author = {Manuel Eberl},
title = {The Hurwitz and Riemann ζ Functions},
journal = {Archive of Formal Proofs},
month = oct,
year = 2017,
note = {\url{http://isaafp.org/entries/Zeta_Function.html},
Formal proof development},
ISSN = {2150914x},
}

License: 
BSD License 
Depends on: 
Bernoulli, Dirichlet_Series, Euler_MacLaurin, Winding_Number_Eval 
Used by: 
Dirichlet_L, Prime_Distribution_Elementary, Prime_Number_Theorem 
Status: [ok] 
This is a development version of this entry. It might change over time
and is not stable. Please refer to release versions for citations.

