
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 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 nonvanishing 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_FunctionAFP,
author = {Manuel Eberl},
title = {The Hurwitz and Riemann ζ Functions},
journal = {Archive of Formal Proofs},
month = oct,
year = 2017,
note = {\url{https://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 
