Functional equation for the zeta function
The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:
Proof
Preparation
There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a light-weighted approach which merely relies on the Fourier series for the first periodic Bernoulli polynomial that
and the Laplace transform identity that
where
A formula for
in ![$-1<\sigma<0$](//latex.artofproblemsolving.com/5/2/1/5211aa0e4af21cc5f79fc5b2c07015ef665a2445.png)
In this article, we will use the common convention that where
. As a result, we say that the original Dirichlet series definition
converges only for
. However, if we were to apply Euler-Maclaurin summation on this definition, we obtain
in which we can extend the ROC of the latter integral to via integration by parts:
When there is
As a result, we obtain a formula for for
:
Expansion of
into Fourier series
In order to go deeper, let's plug
into the previously obtained formula, so that
\[\begin \zeta(s)=s\int_0^\infty\sum_{n=1}^\infty{\sin(2\pi nx)\over n\pi}{\mathrm dx\over x^{s+1}}\] (Error compiling LaTeX. Unknown error_msg)