Analytic Continuation

Beyond the Region of Convergence

The defining series of the Riemann zeta function is

\zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s},

which converges absolutely only when

\operatorname{Re}(s)>1.

At first sight, this seems to restrict the zeta function to the right half-plane. However, many important arithmetic phenomena occur outside this region.

The central idea of analytic continuation is that a function initially defined on one domain may extend uniquely to a much larger domain while preserving analyticity.

Riemann showed that the zeta function extends to a meromorphic function on the entire complex plane, except for a simple pole at

s=1.

This extension is one of the foundational achievements of complex analysis and analytic number theory.

Analytic Functions

A complex function $f(s)$ is analytic on a domain if it admits a convergent power series expansion near every point in the domain.

Analytic functions possess strong rigidity properties. In particular, if two analytic functions agree on an open set, they must agree everywhere on any connected domain where both are defined.

Because of this uniqueness principle, analytic continuation is canonical: there is only one possible continuation.

Failure of the Original Series

The original zeta series diverges when

\operatorname{Re}(s)\leq1.

s=1,

it becomes the harmonic series:

\sum_{n=1}^{\infty}\frac1n,

which diverges.

For

\operatorname{Re}(s)<1,

the terms fail even to approach zero sufficiently rapidly for convergence.

Thus continuation beyond the half-plane

\operatorname{Re}(s)>1

cannot rely on the original definition.

Alternating Zeta Series

A first extension comes from the alternating series

1-\frac1{2^s}+\frac1{3^s}-\frac1{4^s}+\cdots.

This is the Dirichlet eta function:

\eta(s) = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s}.

The series converges for

\operatorname{Re}(s)>0.

Using geometric-series manipulations, one obtains

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

Therefore,

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

This formula analytically continues the zeta function into the larger half-plane

\operatorname{Re}(s)>0,

except at points where the denominator vanishes.

Integral Representation

Another continuation method uses integrals.

For

\operatorname{Re}(s)>1,

one has

\Gamma(s)\zeta(s) = \int_0^\infty \frac{x^{s-1}}{e^x-1}\,dx,

where

\Gamma(s) = \int_0^\infty x^{s-1}e^{-x}\,dx

is the gamma function.

The integral representation allows analytic manipulations that extend beyond the original region of convergence.

Riemann used such methods together with contour integration and Fourier analysis to continue $\zeta(s)$ meromorphically to the whole complex plane.

Pole at $s=1$

The continuation reveals that the zeta function has exactly one singularity:

s=1.

This singularity is a simple pole with residue $1$ . Near $s=1$ ,

\zeta(s) = \frac1{s-1} + \gamma + O(s-1),

where $\gamma$ is the Euler-Mascheroni constant.

The pole reflects the logarithmic growth of the harmonic series and the average density of primes.

Values at Negative Integers

Analytic continuation assigns finite values to the zeta function at points where the original series diverges.

For example,

\zeta(-1)=-\frac1{12},

\zeta(0)=-\frac12,

and more generally,

\zeta(-n) = -\frac{B_{n+1}}{n+1},

where $B_n$ are Bernoulli numbers.

These values arise naturally from the continuation, not from ordinary convergence of the defining series.

Trivial Zeros

The continuation reveals zeros at the negative even integers:

-2,-4,-6,\ldots.

These are called the trivial zeros.

They emerge from the functional equation and the analytic structure of the continuation.

The remaining zeros, called nontrivial zeros, lie in the critical strip

0<\operatorname{Re}(s)<1.

Functional Equation Preview

Analytic continuation alone does not reveal the full symmetry of the zeta function. Riemann discovered the deeper identity

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

which relates values at $s$ and $1-s$ .

This functional equation connects the left and right half-planes and governs the distribution of zeros.

Importance in Number Theory

Analytic continuation transforms the zeta function from a convergent series into a global analytic object.

Without continuation:

the critical strip would be inaccessible,
the functional equation would not exist,
nontrivial zeros could not be studied,
the Prime Number Theorem could not be proved analytically.

The continuation therefore opens the door to the deep relationship between complex analysis and prime numbers.

Modern analytic number theory depends heavily on analogous continuations for many other $L$ -functions and automorphic forms.