\[\Xi(s)\]
\[\Xi(s)=\tfrac12\,s(s-1)\,\pi^{-s/2}\,\Gamma\!\left(\tfrac{s}{2}\right)\zeta(s)\]
The completed xi function is the polished twin of the xi function \(\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)\). Multiplying by the extra factor \(\tfrac12 s(s-1)\) clears the two poles — the pole of \(\Gamma(s/2)\) at \(s=0\) and the pole of \(\zeta(s)\) at \(s=1\) — and turns \(\Xi\) into an entire function, defined and finite everywhere in the complex plane. It keeps the same mirror symmetry \(\Xi(s)=\Xi(1-s)\) about the line \(\mathrm{Re}(s)=\tfrac12\), and takes the tidy boundary values \(\Xi(0)=\Xi(1)=\tfrac12\). This is the form Riemann himself worked with in 1859, and the one on which the Riemann Hypothesis is usually stated : all non-trivial zeros lie on \(\mathrm{Re}(s)=\tfrac12\).
# Completed xi : entire, symmetric, real on the critical line
def Xi(s):
return 1/2 * s*(s-1) * pi^(-s/2) * gamma(s/2) * zeta(s)
print Xi(1/2).n() # 0.497120778188314 (a real minimum)
print Xi(0).n(), Xi(1).n() # 0.5, 0.5
print Xi(0.3).n(), Xi(0.7).n() # equal : Xi(s) = Xi(1-s)
\[\Xi\!\left(\tfrac12\right)=0.4971207781883\ldots\]
On the real axis \(\Xi\) is a real, even function of \(s-\tfrac12\), and its value at the axis of symmetry, \(\Xi\!\left(\tfrac12\right)=0.4971207781\ldots\), is the lowest point of the central valley — the minimum of \(\Xi\) on the real line. Everything in Riemann's construction is organised around this point : it is the fixed point of \(s\mapsto 1-s\), the line the hypothesis singles out, and the height at which the two faces of \(\pi\) — the arithmetic \(\pi\) of \(\zeta(2)=\pi^2/6\) and the geometric \(\pi\) of \(\Gamma\!\left(\tfrac12\right)=\sqrt\pi\) — are seen to meet.
A002410 Nearest integer to the imaginary part of the \(n\)-th zero of \(\Xi\) : 14, 21, 25, 30, 33, ...
\[\Xi(2)=\tfrac12\cdot 2\cdot 1\cdot \pi^{-1}\,\Gamma(1)\,\zeta(2)=\zeta(2)\cdot\pi^{-1}=\frac{\pi}{6}=0.5235987\ldots\]
Here is where the arithmetic side of \(\pi\) walks straight into \(\Xi\). The summatory totient obeys \(\sum_{k\le n}\varphi(k)\sim n^{2}/\bigl(2\,\zeta(2)\bigr)\), so the constant that governs how densely coprime pairs sit among the integers is nothing but \(\zeta(2)=\pi^2/6\). Evaluate \(\Xi\) at \(s=2\) : the clearing factor \(\tfrac12 s(s-1)\) equals \(1\) there, and \(\Gamma(1)=1\), so \(\Xi(2)=\pi^{-1}\zeta(2)=\pi/6\). The density constant of Euler's totient is therefore \(\zeta(2)\) seen inside \(\Xi\) — and by the reflection \(\Xi(2)=\Xi(-1)\) it echoes to the far side of the critical line. The \(\pi\) that the primes assemble through \(\zeta\), and the \(\pi\) that the circle carries through \(\Gamma\), are two readings of the one function \(\Xi\).
A013661 Decimal expansion of \(\zeta(2)=\pi^2/6\) — the totient density constant, entering \(\Xi\) at \(s=2\).
# Where the totient meets Xi : zeta(2) enters at s = 2
def Xi(s):
return 1/2 * s*(s-1) * pi^(-s/2) * gamma(s/2) * zeta(s)
print Xi(2).n() # 0.523598775598299 = pi/6
print (pi/6).n() # 0.523598775598299
print Xi(-1).n() # 0.523598775598299 (reflection Xi(2)=Xi(-1))
# zeta(2) is the density constant of the summatory totient
print (6*zeta(2)).n() # pi^2 -> zeta(2) = pi^2/6