Functional Equation and Analytic Continuation

The functional equation relates values of $\zeta(s)$ and $\zeta(1-s)$ , revealing deep symmetry and enabling analytic continuation beyond the region of convergence.

DefinitionFunctional Equation

The Riemann zeta function satisfies the functional equation:

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

Equivalently, defining $\xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)$ :

\xi(s) = \xi(1-s)

This symmetric form makes the functional equation more transparent.

ExampleDeriving Trivial Zeros

For negative even integers $s = -2n$ where $n \geq 1$ :

\sin\left(\frac{\pi(-2n)}{2}\right) = \sin(-n\pi) = 0

Therefore $\zeta(-2n) = 0$ . These are called trivial zeros.

The non-trivial zeros lie in the critical strip $0 < \Re(s) < 1$ .

DefinitionCritical Strip and Critical Line

The critical strip is $\{s \in \mathbb{C} : 0 \leq \Re(s) \leq 1\}$
The critical line is $\{\Re(s) = 1/2\}$
All non-trivial zeros of $\zeta(s)$ lie in the critical strip
The Riemann Hypothesis conjectures they all lie on the critical line

Remark

The functional equation can be derived using the Poisson summation formula applied to theta functions:

\theta(t) = \sum_{n=-\infty}^{\infty} e^{-\pi n^2 t}

satisfies $\theta(t) = t^{-1/2} \theta(1/t)$ , and integrating against $t^{s/2-1}$ yields the functional equation.

ExampleAnalytic Continuation via Eta Function

The Dirichlet eta function:

\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = \left(1 - 2^{1-s}\right) \zeta(s)

converges for $\Re(s) > 0$ . Since $\eta(s)$ is holomorphic for $\Re(s) > 0$ , we can use:

\zeta(s) = \frac{\eta(s)}{1 - 2^{1-s}}

to continue $\zeta$ to $\Re(s) > 0$ , $s \neq 1$ .

The functional equation reveals that the "information" about $\zeta(s)$ for $\Re(s) < 0$ is completely determined by its values for $\Re(s) > 1$ . This symmetry around $s = 1/2$ is at the heart of the Riemann Hypothesis and connects to deep questions about the distribution of primes.