# 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 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.