Equation 12 — The Hadamard Product for ξ(s): A Deep Dive
$$\xi(s) = \frac{1}{2}\prod_\rho \left(1 - \frac{s}{\rho}\right)$$
1. What the Equation Says
The completed (or "Riemann xi") function is
$$\xi(s) = \tfrac{1}{2}\,s(s-1)\,\pi^{-s/2}\,\Gamma\!\left(\tfrac{s}{2}\right)\zeta(s).$$
This function is entire (the factor s(s−1) kills the pole of ζ at s = 1 and the pole of Γ(s/2) at s = 0), satisfies the perfect functional equation ξ(s) = ξ(1−s), and its zeros are exactly the nontrivial zeros ρ of ζ(s) — the trivial zeros at −2, −4, −6, … are absorbed by Γ(s/2).
The Hadamard product then says: ξ is completely reconstructible from its zeros alone, just as a polynomial is determined (up to a constant) by its roots:
$$\xi(s) = \xi(0)\prod_\rho\left(1-\frac{s}{\rho}\right) = \frac{1}{2}\prod_\rho\left(1-\frac{s}{\rho}\right),$$
with the product taken symmetrically, i.e. as $\lim_{T\to\infty}\prod_{|\operatorname{Im}\rho| A beautiful consequence of the pairing ρ ↔ 1−ρ: grouping paired factors gives $$\left(1-\frac{s}{\rho}\right)\left(1-\frac{s}{1-\rho}\right) = 1 - \frac{s(1-s)}{\rho(1-\rho)},$$ which converges absolutely because $\sum_\rho 1/|\rho(1-\rho)| \asymp \sum_\gamma (\log\gamma)/\gamma^2 < \infty$. This is the "pair-and-drop" trick of Step 5 made concrete. Bernhard Riemann (1859). In his single 8-page paper on number theory, "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Primes Less Than a Given Magnitude"), Riemann asserted this product formula for his ξ function (he wrote it as a function Ξ(t) with s = ½ + it) with essentially no proof. He needed it as the engine of his explicit formula connecting primes to zeros. Riemann wrote that the product representation follows because the number of roots grows suitably slowly — a claim that was true but far beyond the analysis of 1859. Jacques Hadamard (1893). The product remained unproven for 34 years. Hadamard, in his prize-winning memoir "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann" (Journal de Mathématiques Pures et Appliquées, 4e série, t. 9, 1893, pp. 171–215), created the general theory of entire functions of finite order/genus — what we now call the Hadamard factorization theorem — precisely to justify Riemann's claim. This memoir won the Grand Prix des Sciences Mathématiques of the French Academy (1892 competition). It is often said that the entire modern theory of entire functions was born as a lemma for the Riemann Hypothesis. Hans von Mangoldt (1895). Used Hadamard's product to give the first rigorous proof of Riemann's explicit formula, in "Zu Riemanns Abhandlung 'Über die Anzahl der Primzahlen unter einer gegebenen Grösse'" (J. reine angew. Math. 114, 1895). Hadamard & de la Vallée Poussin (1896). Both proofs of the Prime Number Theorem — obtained independently that year — run directly through this product and its corollary for ζ′/ζ. De la Vallée Poussin's classical zero-free region also comes from the partial-fraction formula in Step 6. Xian-Jin Li (1997). The constant B = ξ′(0)/ξ(0) = log 2π − 1 − γ/2 ≈ −0.0230957 reappeared a century later: Li proved that RH is equivalent to the positivity of the coefficients λₙ = Σ_ρ [1 − (1 − 1/ρ)ⁿ], and λ₁ = −B — the connection noted in your Step 4. Order of ξ. One shows |ξ(s)| ≪ exp(C|s| log|s|), so ξ has order exactly 1 (via Stirling for Γ and elementary bounds for ζ). Crucially, ξ is order 1 but not of exponential type in the naive sense — Σ 1/|ρ| diverges (there are ~ (T/2π) log T zeros up to height T, so Σ 1/γ ~ (log T)²/4π diverges). This forces the genus-1 Weierstrass factors e^{s/ρ}. Hadamard factorization gives ξ(s) = e^{A+Bs} ∏(1 − s/ρ)e^{s/ρ}. A: ξ(0) = ξ(1) = ½ · 1 · π^{−1/2} Γ(½) ζ(... ) → careful limit gives ξ(0) = ½, so A = −log 2. B: The symmetry ξ(s) = ξ(1−s) forces ξ′/ξ(s) = −ξ′/ξ(1−s). Computing ξ′(0)/ξ(0) from the definition (using ζ(0) = −½, ζ′(0) = −½ log 2π, and the digamma value ψ(1) = −γ) yields
$$B = \log 2\pi - 1 - \tfrac{\gamma}{2}.$$
Equivalently, B = −Σ_ρ Re(1/ρ) (paired) — a fact used constantly in analytic number theory. Note Re(1/ρ) > 0 for every zero, so B < 0. Symmetric pairing absorbs e^{Bs} and all e^{s/ρ}, leaving the clean product. Corollary (Step 6). Taking the logarithmic derivative of both the definition of ξ and of the Hadamard product and solving for ζ′/ζ:
$$\frac{\zeta'(s)}{\zeta(s)} = B + \frac{1}{2}\log\pi - \frac{1}{s-1} - \frac{1}{2}\frac{\Gamma'}{\Gamma}\!\left(\frac{s}{2}+1\right) + \sum_\rho\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right).$$
This is the identity behind every explicit formula (Riemann–von Mangoldt, Weil), every zero-free region, and every zero-density theorem. It says: the logarithmic derivative of ζ is a sum of simple poles at the zeros, plus explicitly known "archimedean" terms. (Caveat: journal-archive URLs occasionally move; the DOIs above are stable.) All examples use Zeros are paired ρ ↔ 1−ρ (numerically, 1−ρ = ρ̄ on the critical line), using the identity (1−s/ρ)(1−s/(1−ρ)) = 1 − s(1−s)/(ρ(1−ρ)): Typical output shows the relative error shrinking roughly like (log T)/T — the product converges, but slowly (tail of Σ 1/γ²). This slow convergence is itself instructive: it reflects the density (T/2π) log T of zeros. Here the summand behaves like s/(ρ(s−ρ)) ~ 1/γ², so convergence is absolute; with N = 500 zeros you already reproduce ζ′(3)/ζ(3) = −0.19812624288563685… to several digits. Each term 2 Re(1/ρ) is positive if and only if Re ρ > 0 in the right way — and Li's criterion says RH ⇔ λₙ > 0 for all n. The Hadamard product is what makes the λₙ well-defined in the first place.
2. History
3. Sketch of the Full Derivation
4. References and Links
5. Computational Examples in Python
mpmath (pip install mpmath).5.1 Verify the constant B = log 2π − 1 − γ/2 = ξ′(0)/ξ(0)
from mpmath import mp, mpc, mpf, zeta, gamma, pi, log, euler, diff, zetazero, re
mp.dps = 30
def xi(s):
s = mpc(s)
return mpf('0.5') * s * (s - 1) * pi**(-s/2) * gamma(s/2) * zeta(s)
B_closed = log(2*pi) - 1 - euler/2
B_numeric = diff(xi, 0) / xi(0)
print("closed form :", B_closed)
print("xi'(0)/xi(0):", B_numeric)
# Both print -0.0230957089661210338...
5.2 The Hadamard product, truncated over the first N zeros
from mpmath import mp, mpc, mpf, zetazero
mp.dps = 25
def xi_hadamard(s, N):
"""Truncated symmetric Hadamard product over first N zero-pairs."""
s = mpc(s)
prod = mpf('0.5') # xi(0) = 1/2
for n in range(1, N + 1):
rho = zetazero(n) # zero with Im > 0
prod *= 1 - s*(1 - s) / (rho*(1 - rho))
return prod
s_test = mpc('2.0')
exact = xi(s_test)
print("exact xi(2) =", exact)
for N in (10, 50, 200):
approx = xi_hadamard(s_test, N)
print(f"N={N:4d}: product = {approx}, rel.err = {abs(approx-exact)/abs(exact)}")
5.3 Verify the key corollary (Step 6) for ζ′/ζ
from mpmath import mp, mpc, zetazero, digamma, zeta, log, pi, euler
mp.dps = 20
def zeta_logderiv_from_zeros(s, N):
s = mpc(s)
B = log(2*pi) - 1 - euler/2
total = B + log(pi)/2 - 1/(s - 1) - digamma(s/2 + 1)/2
for n in range(1, N + 1):
rho = zetazero(n)
for r in (rho, rho.conjugate()): # include lower-half-plane zeros
total += 1/(s - r) + 1/r
return total
s_test = mpc('3.0')
exact = zeta(s_test, derivative=1) / zeta(s_test)
print("exact zeta'/zeta(3) :", exact)
for N in (10, 100, 500):
print(f"N={N:4d} via zeros :", zeta_logderiv_from_zeros(s_test, N))
5.4 Bonus — see B = −λ₁ (Li's first coefficient)
lam1 = 0
for n in range(1, 2001):
rho = zetazero(n)
lam1 += re(1/rho + 1/rho.conjugate()) # paired: 2*Re(1/rho)
print("partial lambda_1 :", lam1) # converges (slowly) to +0.0230957...
print(" -B :", -(log(2*pi) - 1 - euler/2))
6. Why This Equation Matters for Your Paper