The series $\zeta(s) = \sum 1/n^s$ only converges for $s > 1$. But the zeta function “exists” for all complex $s \neq 1$. How?

Analytic continuation is the process of extending a function beyond its original domain of definition — and the extension, when it exists, is unique. The zeta function defined by the series for $s > 1$ has exactly one smooth extension to the rest of the complex plane.

The key idea

An analytic function (also called holomorphic) is one that can be expressed as a convergent power series in a neighbourhood of every point. Analytic functions are extraordinarily rigid — if you know the function on any region (even a tiny interval), the rest is completely determined.

Think of it like a jigsaw puzzle where every piece is uniquely shaped: once you place a few pieces, the rest can only go one way. If two analytic functions agree on any interval, they agree everywhere they’re both defined.

This rigidity means: if a series defines an analytic function on part of the complex plane, there is at most one way to extend it to a larger domain while keeping it analytic.

Extending the zeta function

For the zeta function, the extension happens in stages.

Step 1: Start with $\zeta(s) = \sum 1/n^s$ for $\text{Re}(s) > 1$.

Step 2: The alternating series $\eta(s) = 1 - 1/2^s + 1/3^s - 1/4^s + \cdots$ converges for $\text{Re}(s) > 0$. A simple identity relates them:

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

This gives $\zeta(s)$ for $\text{Re}(s) > 0$ (except where $2^{1-s} = 1$).

Step 3: The functional equation

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

relates $\zeta(s)$ to $\zeta(1-s)$. Since the gamma function is defined everywhere (except at poles), this extends $\zeta$ to the entire complex plane.

The result: $\zeta(s)$ is defined for all $s \in \mathbb{C}$ except $s = 1$, where it has a simple pole.

What the extended zeta function reveals

The series $\sum 1/n^s$ says nothing about $s$ with $\text{Re}(s) < 1$. But the analytically continued function has rich structure there:

Trivial zeros: $\zeta(-2) = 0, \zeta(-4) = 0, \zeta(-6) = 0, \ldots$ These come from the poles of the gamma function.

Non-trivial zeros: Infinitely many zeros in the critical strip $0 < \text{Re}(s) < 1$. The Riemann Hypothesis says they all lie on $\text{Re}(s) = 1/2$.

Values at negative integers: $\zeta(0) = -1/2$, $\zeta(-1) = -1/12$, $\zeta(-2) = 0$. The famous “$1 + 2 + 3 + \cdots = -1/12$” is a (misleading) shorthand for $\zeta(-1) = -1/12$ — the value of the analytically continued function, not the divergent series.

The critical strip is where the action is: The zeros of $\zeta$ in this strip control the error term in the Prime Number Theorem. The further the zeros are from the line $\text{Re}(s) = 1$, the more regularly the primes are distributed. RH pushes all zeros to $\text{Re}(s) = 1/2$ — the maximum possible regularity.

Beyond zeta

Analytic continuation isn’t special to the zeta function. It works for:

• L-functions: Dirichlet L-functions, Artin L-functions, Hasse-Weil L-functions of elliptic curves — all have analytic continuations with functional equations, and the location of their zeros encodes deep arithmetic information.
• The gamma function: Defined by the integral for $\text{Re}(s) > 0$, extended to all $\mathbb{C}$ by the functional equation $\Gamma(s+1) = s\Gamma(s)$.
• Modular forms: Their L-functions have analytic continuations that connect number theory to geometry.

The pattern: a function defined by a series or integral in some region extends uniquely to a larger domain, and the extended function reveals structure invisible in the original definition.