\[\xi(s)\]
\[\xi(s)=\pi^{-s/2}\,\Gamma\!\left(\tfrac{s}{2}\right)\zeta(s)\qquad\Longrightarrow\qquad \xi(s)=\xi(1-s)\]
The completed zeta function \(\xi\) is the object that reveals the hidden symmetry of \(\zeta\). On its own the Riemann zeta function has no obvious symmetry, but multiplying it by the factor \(\pi^{-s/2}\Gamma(s/2)\) — the Gamma factor — produces a function that is perfectly mirror-symmetric about the line \(\mathrm{Re}(s)=\tfrac12\) : \(\xi(s)=\xi(1-s)\). In other words, \(\Gamma\) is precisely the piece that \(\zeta\) is missing to become symmetric. The two are not separate functions that happen to share \(\pi\) — they are the two halves of a single object.
A note on convention. Two closely related functions carry the name \(\xi\). The form used on this page, \(\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)\), is the cleaner one for reading off the symmetry \(\xi(s)=\xi(1-s)\), and it is where \(\Gamma(1/4)\) surfaces directly at \(s=\tfrac12\). Some authors reserve \(\xi\) for Riemann's completed form \(\tfrac12\,s(s-1)\,\pi^{-s/2}\Gamma(s/2)\zeta(s)\), whose extra factor \(\tfrac12 s(s-1)\) cancels the poles at \(s=0\) and \(s=1\) and makes \(\xi\) an entire function with \(\xi(0)=\xi(1)=\tfrac12\) and \(\xi(\tfrac12)=0.4971207781\ldots\). Both obey the same reflection \(\xi(s)=\xi(1-s)\) ; the two differ only by that pole-clearing factor, and the meeting of the two faces of \(\pi\) at \(s=\tfrac12\) is the same in either.
\[\zeta(s)=2^{s}\,\pi^{s-1}\sin\!\left(\tfrac{\pi s}{2}\right)\Gamma(1-s)\,\zeta(1-s)\]
This is Riemann's functional equation (1859), the analytic bridge that ties \(\zeta\) and \(\Gamma\) together and, through them, the two very different worlds in which \(\pi\) appears. On one side, \(\zeta\) reaches the primes through the Euler product \(\zeta(s)=\prod_p (1-p^{-s})^{-1}\) ; on the other, \(\Gamma\) and \(\sin\) carry the geometry of the circle. The equation is the statement that these are reflections of one another across \(\mathrm{Re}(s)=\tfrac12\). The symmetric form \(\xi(s)=\xi(1-s)\) is simply this same law, cleaned of its poles by the Gamma factor.
# The completed zeta function is symmetric : xi(s) = xi(1-s)
def xi(s):
return pi^(-s/2) * gamma(s/2) * zeta(s)
print xi(0.3).n(), xi(0.7).n() # equal
print xi(2.0).n(), xi(-1.0).n() # equal
# Riemann's functional equation reproduces zeta(s) from zeta(1-s)
s = 3
lhs = zeta(s)
rhs = 2^s * pi^(s-1) * sin(pi*s/2) * gamma(1-s) * zeta(1-s)
print (lhs - rhs).n() # 0
\[\xi\!\left(\tfrac12\right)=\pi^{-1/4}\,\Gamma\!\left(\tfrac14\right)\zeta\!\left(\tfrac12\right)=-3.9769662255\ldots\]
The point \(s=\tfrac12\) is the axis of the symmetry — the fixed point of \(s\mapsto 1-s\), and the line on which the Riemann Hypothesis places every non-trivial zero. It is the one place where the argument of \(\Gamma\) (namely \(s/2=\tfrac14\)) and the argument of \(\zeta\) (namely \(\tfrac12\)) are both remarkable : \(\Gamma\!\left(\tfrac14\right)\) is the gateway to the lemniscate, and \(\zeta\!\left(\tfrac12\right)\) sits at the heart of the hypothesis. Here the two faces of \(\pi\) — the arithmetic one born from \(\zeta(2)=\pi^2/6\) and the primes, and the geometric one born from \(\Gamma\!\left(\tfrac12\right)=\sqrt{\pi}\) and the circle — meet. They are the same number seen at two heights of the same function.
A068466 Decimal expansion of \(\Gamma\!\left(\tfrac14\right)\) — transcendental, gateway to the lemniscate.
A059750 Decimal expansion of \(\zeta\!\left(\tfrac12\right)=-1.4603545088\ldots\)
A062539 Decimal expansion of the lemniscate constant \(\varpi=\Gamma\!\left(\tfrac14\right)^2/(2\sqrt{2\pi})\).
# xi at the critical point s = 1/2
val = pi^(-1/4) * gamma(1/4) * zeta(1/2)
print val.n() # -3.97696622550651...
# The two faces of pi meet here :
print (gamma(1/2)^2).n() # pi (geometry : Gamma(1/2)=sqrt(pi))
print (6*zeta(2)).n() # pi^2 (arithmetic : zeta(2)=pi^2/6)
# Gamma(1/4) opens onto the lemniscate constant
varpi = gamma(1/4)^2 / (2*sqrt(2*pi))
print varpi.n() # 2.62205755429211...
\[\varpi=\frac{\Gamma\!\left(\tfrac14\right)^{2}}{2\sqrt{2\pi}}=\frac{\pi}{M\!\left(1,\sqrt{2}\right)}=2.6220575542\ldots\]
The constant \(\Gamma\!\left(\tfrac14\right)\) that surfaces in \(\xi\!\left(\tfrac12\right)\) is itself the doorway to the lemniscate constant \(\varpi\), which is to the lemniscate (the figure-eight curve) exactly what \(\pi\) is to the circle : the ratio of the curve's perimeter to its diameter. It also equals \(\pi/M(1,\sqrt2)\), where \(M\) is Gauss's arithmetic–geometric mean — so the same \(\pi\) that the primes build through \(\zeta\) reappears here, divided by Gauss's mean of \(1\) and \(\sqrt2\). The reflection formula \(\Gamma\!\left(\tfrac14\right)\Gamma\!\left(\tfrac34\right)=\pi\sqrt2\) closes the circle : whichever way one turns, \(\pi\) is waiting. Like \(\pi\), \(\varpi\) is transcendental (Siegel 1932, Schneider 1937).
A tale of two constants. Just as the circle carries both a “radius” constant \(\pi\) and a “perimeter” constant \(2\pi\), the lemniscate comes in a pair that differ by a factor of two — and, as with \(h\) and \(\hbar\) in physics, authors disagree on which deserves the bare name. The value \(\varpi=2.6220575\ldots\) (A062539) is the half-arc, while the full arc length of the lemniscate is \(2\varpi=\Gamma\!\left(\tfrac14\right)^{2}/\sqrt{2\pi}=5.2441151\ldots\) (A064853). Both are the same \(\Gamma\!\left(\tfrac14\right)^{2}\) up to the placement of the factor \(\sqrt2\) — the very \(\sqrt2\) that runs through Gauss's mean \(M(1,\sqrt2)\).
A064853 Decimal expansion of the full lemniscate arc length \(2\varpi=\Gamma\!\left(\tfrac14\right)^2/\sqrt{2\pi}\).
# The lemniscate constant, three equal faces
print (gamma(1/4)^2 / (2*sqrt(2*pi))).n() # 2.62205755429212
print (pi / agm(1, sqrt(2))).n() # 2.62205755429212
# Euler's reflection at s = 1/4 : Gamma(1/4) Gamma(3/4) = pi sqrt(2)
print (gamma(1/4)*gamma(3/4)).n() # 4.44288293816...
print (pi*sqrt(2)).n() # 4.44288293816...