Functional equation of the Riemann zeta function

Remarks

One proof uses the theta function $\theta(t) = \sum_{n \in \Z} e^{-\pi n^2 t}$ for $t > 0$. Poisson summation gives $\theta(1/t) = \sqrt{t} \, \theta(t)$. A Mellin transform expresses $\xi_0(s)$ as an integral of $(\theta(t) - 1)/2$ against $t^{s/2 - 1}$. Splitting the integral at $t = 1$ and using the symmetry of $\theta$ gives a formula that is visibly invariant under $s \mapsto 1 - s$. The same formula gives the meromorphic continuation of $\zeta$ to $\C$. Example: at $s = -1$ the right side is $\frac{1}{2} \cdot \frac{1}{\pi^2} \cdot (-1) \cdot \Gamma(2) \cdot \zeta(2) = -\frac{1}{12}$, using $\zeta(2) = \pi^2/6$, so $\zeta(-1) = -1/12$. At $s = -2, -4, -6, \dots$ the sine factor vanishes while $\Gamma(1 - s) \zeta(1 - s)$ is finite and nonzero, so these are zeros of $\zeta$, the trivial zeros.

Going deeper

For a student

The equation shows that the zeros of $\zeta$ with $\operatorname{Re} s < 0$ are the trivial zeros $-2, -4, \dots$. The other zeros lie in the critical strip $0 \le \operatorname{Re} s \le 1$, placed symmetrically about the line $\operatorname{Re} s = 1/2$. The Riemann hypothesis says that they lie on this line. Dirichlet L-functions of primitive characters and many other L-functions satisfy functional equations of the same shape.