Functional equation for the zeta function

The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:

$\zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s)$

Proof

Two useful identities

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

$B_1(x)\triangleq\{x\}-\frac12=-\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n}$

A formula for $\zeta(s)$ in $-1<\sigma<0$

In this article, we will use the common convention that $s=\sigma+it$ where $\sigma,t\in\mathbb R$ . As a result, we say that the original Dirichlet series definition $\zeta(s)\triangleq\sum_{k=1}^\infty{1\over k^s}$ converges only for $\sigma>1$ . However, if we were to apply Euler-Maclaurin summation on this definition, we obtain

$\zeta(s)=\frac12+{s\over s-1}-s\int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx$

in which we can extend the ROC of the latter integral to $\sigma>-1$ via integration by parts:

$\begin{align*} \int_1^\infty{B_1(x)\over x^{s+1}}\mathrm dx &=\left.{B_2(x)\over2x^{s+1}}\right|_1^\infty+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx \\ &={B_2\over2}+{s+1\over2}\int_1^\infty{B_2(x)\over x^{s+2}}\mathrm dx \end{align*}$

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Functional equation for the zeta function

Proof

Two useful identities

A formula for $\zeta(s)$ in $-1<\sigma<0$

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Functional equation for the zeta function

Proof

Two useful identities

A formula for in

A formula for $\zeta(s)$ in $-1<\sigma<0$