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The
Hurwitz
and
Riemann
ζ
Functions
Title: |
The Hurwitz and Riemann ζ Functions |
Author:
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Manuel Eberl
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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},
}
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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.
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