Section 6.1 The functional equation for \(\zeta\)
A “random” Dirichlet series \(\sum_n a_n n^{-s}\) will not exhibit very interesting analytic behavior beyond its abscissa of absolute convergence. However, we already know that \(\zeta\) is atypical in this regard, in that we can extend it at least as far as \(\Real(s) > 0\) if we allow the simple pole at \(s=1\text{.}\) One of Riemann's key observations is that in the strip \(0 \lt \Real(s) \lt 1\text{,}\) \(\zeta\) obeys a symmetry property relating \(\zeta(s)\) to \(\zeta(1-s)\text{;}\) once we prove this, we will then be able to extend \(\zeta\) all the way across the complex plane. (This is essentially Riemann's original proof; several others are possible.)
We first recall the definition and basic properties of the \(\Gamma\) function.
Definition 6.1.
We define
\begin{equation*}
\Gamma(s) := \int_0^\infty e^{-t} t^{s-1}\,dt
\end{equation*}
for \(\Real(s) > 0\text{.}\) Using integration by parts, one may check that
\begin{equation}
\Gamma(s+1) = s \Gamma(s) \qquad (\Real(s) > 0).\tag{6.1.1}
\end{equation}
Using (6.1.1), we may extend \(\Gamma\) to a meromorphic function on all of \(\CC\text{,}\) with simple poles at \(s=0,-1,\dots\text{.}\) Since \(\Gamma(1) = \int_0^\infty e^{-t}\,dt = 1\text{,}\) we have that for \(n\) a nonnegative integer,
\begin{equation*}
\Gamma(n+1) = n!;
\end{equation*}
that is, \(\Gamma\) provides a meromorphic extension of the factorial function over \(\CC\text{.}\)
Substituting \(t = \pi n^2 x\) in the definition of \(\Gamma\text{,}\) we have
\begin{equation}
\pi^{-s/2} \Gamma(s/2) n^{-s} = \int_0^\infty x^{s/2 - 1} e^{-n^2 \pi x}\,dx
\qquad \Real(s) > 0.\tag{6.1.2}
\end{equation}
If we sum over \(n\text{,}\) we can interchange the sum and integral for \(\Real(s) > 1\) because the sum-integral converges absolutely. Hence
\begin{equation*}
\pi^{-s/2} \Gamma(s/2) \zeta(s)
= \int_0^\infty x^{s/2 - 1} \omega(x) \,dx
\qquad \Real(s) > 1
\end{equation*}
for
\begin{equation*}
\omega(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}.
\end{equation*}
It is slightly more convenient to work with the function \(\theta\) defined by
\begin{equation*}
\theta(x) := \sum_{n=-\infty}^\infty e^{-n^2 \pi x},
\end{equation*}
which clearly satisfies \(2 \omega(x) = \theta(x) - 1\text{.}\)
At this point, Riemann recognized \(\theta\) as a function of the sort considered by Jacobi in the late 19th century. From that work, Riemann knew about the identity
\begin{equation}
\theta(x^{-1}) = x^{1/2} \theta(x) \qquad (x > 0).\tag{6.1.3}
\end{equation}
Postponing the proof to
Corollary 6.7, let us see now how to use
(6.1.3) to get a functional equation for
\(\zeta\text{.}\)
Returning to
\begin{equation*}
\pi^{-s/2} \Gamma(s/2) \zeta(s) = \int_0^\infty x^{s/2 - 1} \omega(x) \,dx,
\end{equation*}
we take the natural step of splitting the integral at \(x=1\text{,}\) then substituting \(1/x\) for \(x\) in the integral from 0 to 1. This yields
\begin{equation*}
\pi^{-s/2} \Gamma(s/2) \zeta(s) = \int_1^\infty x^{s/2-1} \omega(x)\,dx
+ \int_1^\infty x^{-s/2-1} \omega(1/x)\,dx.
\end{equation*}
\begin{equation*}
\omega(x^{-1}) = -\frac{1}{2} + \frac{1}{2}x^{1/2} + x^{1/2} \omega(x),
\end{equation*}
yielding
\begin{equation*}
\int_1^\infty x^{-s/2-1} \omega(x^{-1})\,dx =
-\frac{1}{s} + \frac{1}{s-1} + \int_1^\infty x^{-s/2-1/2}
\omega(x)\,dx.
\end{equation*}
Hence for \(\Real(s) > 1\text{,}\)
\begin{equation}
\pi^{-s/2} \Gamma(s/2) \zeta(s) = -\frac{1}{s(1-s)} + \int_1^\infty (x^{s/2-1} + x^{(1-s)/2-1}) \omega(x)\,dx.\tag{6.1.4}
\end{equation}
Now observe that the left side of
(6.1.4) represents a meromorphic function on
\(\Real(s) > 0\text{,}\) whereas the right side of
(6.1.4) represents a meromorphic function on all of
\(\CC\text{,}\) because the integral converges absolutely for all
\(z\text{.}\) (That's because
\(\omega(x) = O(e^{-\pi x})\) as
\(x \to +\infty\text{.}\)) This has tons of consequences.
