Different sources appear to give different formulas for Riemann's famous functional equation for the zeta function. Specifically, Wikipedia (here) gives
$$\zeta(s) = 2^s \pi ^{s-1} 2 \sin \left(\frac{s \pi }{2}\right) \Gamma (1-s) \zeta (1-s)$$
whereas "Riemann's Zeta Function" by H.M. Edwards gives the formula
$$\zeta(s) = \Pi (-s) (2 \pi )^{s-1} 2 \sin \left(\frac{s \pi }{2}\right) \zeta (1-s)$$
His $\Pi$ notation is the original form of the gamma function, where $\Pi(s) = \Gamma(s-1)$. Thus $\Pi (-s) = \Gamma(-s-1)$, and so Edwards' formula is
$$
\begin{aligned}
\zeta(s) &= \Gamma (-s-1) (2 \pi )^{s-1} 2 \sin \left(\frac{s \pi }{2}\right) \zeta (1-s)
\\&= 2^s \pi ^{s-1} \sin \left(\frac{\pi s}{2}\right) \Gamma (-s-1) \zeta (1-s)
\end{aligned}
$$
So, Wikipedia has $\Gamma (1-s)$ as a factor, where Edwards has $\Gamma (-s-1)$ – and only one of them can be right.
Trouble is, calculation using Mathematica suggests that neither of them are right! Using real values of $s$ in the range $s = [-1.5, 1.5]$ produces the following table:
What I have done here is use Mathematica's values for $\zeta(1-s)$, and plugged them into the two variants of the functional equation.
Clearly, I am doing something very badly wrong! There are two issues:
- Why the disparity between Edwards and Wikipedia? Perhaps I am wrong to write $\Pi (-s) = \Gamma(-s-1)$?
- Why does Mathematica disagree with the published versions on the functional equation?
It's clear to me that I must have made some very bad assumptions – but what are they?
For those who wish to replicate the table, the code is
Grid[Prepend[Table[{NumberForm[N[s], 3], NumberForm[N[Zeta[s]], 3],
NumberForm[
N[Gamma[-s - 1]*(2*Pi)^(s - 1)*2*Sin[(s*Pi)/2]*Zeta[1 - s]],
3],
NumberForm[
N[2^s*Pi^(s - 1)*2*Sin[(s*Pi)/2]*Gamma[1 - s]*Zeta[1 - s]],
3]},
{s, -(3/2), 3/2,
1/4}], {Column[{"", TraditionalForm[s], "", ""}],
Column[{"", TraditionalForm[Zeta[s]], "(Mathematica)", ""}],
Column[{"",
TraditionalForm[
HoldForm[2^s*Pi^(s - 1)*Sin[(s*Pi)/2]*Gamma[-s - 1]*
Zeta[1 - s]]], "(Edwards)", ""}],
Column[{"",
TraditionalForm[
HoldForm[2^s*Pi^(s - 1)*2*Sin[(s*Pi)/2]*Gamma[1 - s]*
Zeta[1 - s]]], "(Wikipedia)", ""}]}],
Alignment -> Left, Spacings -> {1, 0.5},
Dividers -> {{Thick, Thin, Thin, Thin, Thick}, {Thick, True, Thin,
Thin, Thin, Thin, Thin,
Thin, Thin, Thin, Thin, Thin, Thin, Thin, Thick}}]
Best Answer
Note that $\Pi(s) = \Gamma(s+1)$ (easy check $\Pi(n)=n!, \Gamma(n)=(n-1)!$) so the two expressions in Edwards and Wikipedia match.
As for the computations, the third column has an extra $2$ factor (it has both $2^s$ and $2$ and it's either $2^s(\pi)^{s-1}$ or $2(2\pi)^{s-1}$, last one to make a homogeneous constant raised to same power $s-1$), and then indeed clearly the first column and the third column would then match (the third column being twice the first column)