Functional Equation of Zeta

The functional equation of the Riemann zeta function is one of the most beautiful identities in mathematics, revealing symmetry and enabling analytic continuation.

TheoremFunctional Equation

The Riemann zeta function satisfies:

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

for all $s \in \mathbb{C}$ except $s = 0, 1$ .

Equivalently, defining the completed zeta function:

\xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)

we have the symmetric form:

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

ExampleChecking at Special Points

At $s = 0$ :

LHS: $\zeta(0) = -1/2$
RHS: $2^0 \pi^{-1} \sin(0) \Gamma(1) \zeta(1)$ involves $\sin(0) = 0$ and the pole of $\zeta(1)$

The functional equation must be interpreted carefully at special points using limits.

For $s = 1/2$ :

\zeta(1/2) = 2^{1/2} \pi^{-1/2} \sin(\pi/4) \Gamma(1/2) \zeta(1/2)

Checking: $\sin(\pi/4) = 1/\sqrt{2}$ and $\Gamma(1/2) = \sqrt{\pi}$ gives identity (after algebraic simplification).

Remark

The functional equation has several important consequences:

Analytic continuation: Extends $\zeta(s)$ from $\Re(s) > 1$ to all $\mathbb{C} \setminus \{1\}$
Trivial zeros: Shows $\zeta(-2n) = 0$ for $n = 1, 2, 3, \ldots$
Critical strip: Non-trivial zeros lie in $0 < \Re(s) < 1$
Symmetry: If $\rho$ is a zero, so is $1-\rho$ and $\bar{\rho}$

ExampleComputing Negative Values

For $s = -1$ :

\zeta(-1) = 2^{-1} \pi^{-2} \sin(-\pi/2) \Gamma(2) \zeta(2)

= \frac{1}{2\pi^2} \cdot (-1) \cdot 1 \cdot \frac{\pi^2}{6} = -\frac{1}{12}

This gives the famous regularization $1 + 2 + 3 + 4 + \cdots = -1/12$ (in the zeta function sense).

DefinitionCompleted Zeta Function

The completed zeta function $\xi(s)$ removes the pole and trivial zeros:

\xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)

Properties:

$\xi(s)$ is entire (holomorphic on all $\mathbb{C}$ )
$\xi(s) = \xi(1-s)$ (perfect symmetry)
Zeros of $\xi$ are exactly the non-trivial zeros of $\zeta$
Real on the critical line $\Re(s) = 1/2$

The functional equation is fundamental to all deep work on the zeta function, providing both analytic continuation and the symmetry underlying the Riemann Hypothesis.