Riemann’s functional equation for $\zeta(s)$: what am I doing wrong

complex-analysisgamma functionriemann-zeta

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:

enter image description here

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:

  1. Why the disparity between Edwards and Wikipedia? Perhaps I am wrong to write $\Pi (-s) = \Gamma(-s-1)$?
  2. 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)

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