Equation 45: ζ(3) — Apéry's Constant and the Irrationality Proof

1. Overview

The value

$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} = 1.2020569031595942854\ldots$$

is known as Apéry's constant. Unlike the even zeta values, where Euler (1735) proved the beautiful closed forms

$$\zeta(2) = \frac{\pi^2}{6}, \qquad \zeta(4) = \frac{\pi^4}{90}, \qquad \zeta(2k) = \frac{(-1)^{k+1} B_{2k} (2\pi)^{2k}}{2(2k)!},$$

no closed form is known for any odd zeta value ζ(3), ζ(5), ζ(7), … This asymmetry is one of the deepest unsolved mysteries connected to the Riemann zeta function. The structural reason: Euler's proof for even values exploits the functional equation ζ(1−s) = 2(2π)⁻ˢ cos(πs/2)Γ(s)ζ(s), but at odd positive integers the cosine factor vanishes, so the functional equation gives no information — it only tells us ζ(−2k) = 0 (the trivial zeros of the Riemann Hypothesis!).

2. History: Roger Apéry and the 1978 Bombshell

Roger Apéry (1916–1994), a French mathematician at the University of Caen, announced in June 1978 at the Journées Arithmétiques de Marseille-Luminy — at age 61, virtually unknown for research — that ζ(3) is irrational. The announcement was met with widespread disbelief. His lecture, delivered in a chaotic style with gaps and cryptic assertions ("they grow on trees in my garden," he reportedly quipped when asked where a key identity came from), was initially dismissed by many in the audience.

Within two months, Henri Cohen, Hendrik Lenstra, and Alfred van der Poorten verified every step. Van der Poorten's famous exposé:

A. van der Poorten, "A proof that Euler missed... Apéry's proof of the irrationality of ζ(3)," Mathematical Intelligencer 1 (1979), 195–203. 📄 Link: https://www.ega-math.narod.ru/Apery1.htm (full text) and https://doi.org/10.1007/BF03028234

Apéry's own terse paper:

R. Apéry, "Irrationalité de ζ(2) et ζ(3)," Astérisque 61 (1979), 11–13. 📄 Link: http://www.numdam.org/item/AST_1979__61__11_0/

3. The Machinery of the Proof

3.1 The Central-Binomial Series

Apéry's starting point is the remarkable fast-converging series:

$$\zeta(3) = \frac{5}{2} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 \binom{2n}{n}}$$

This identity was actually known earlier (it appears in work related to Markov, 1890, and was rediscovered several times), but Apéry was the first to exploit it arithmetically.

3.2 The Apéry Recurrence

Define two sequences satisfying the same three-term recurrence:

$$n^3 u_n = (34n^3 - 51n^2 + 27n - 5)\, u_{n-1} - (n-1)^3\, u_{n-2}$$

Note that 34n³ − 51n² + 27n − 5 = P(n) is sometimes written as (2n−1)(17n² −17n + 5). Two solutions:

• aₙ (the "Apéry numbers") with a₀ = 1, a₁ = 5: $$a_n = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{n+k}{k}^2$$
• bₙ with b₀ = 0, b₁ = 6: $$b_n = \sum_{k=0}^{n} \binom{n}{k}^2 \binom{n+k}{k}^2 \left( \sum_{m=1}^{n} \frac{1}{m^3} + \sum_{m=1}^{k} \frac{(-1)^{m-1}}{2m^3 \binom{n}{m}\binom{n+m}{m}} \right)$$

3.3 Why This Proves Irrationality

1. Miraculous integrality: aₙ ∈ ℤ, and 2·lcm(1,…,n)³ · bₙ ∈ ℤ — highly nonobvious, since the recurrence divides by n³.
2. Fast convergence: bₙ/aₙ → ζ(3), with

$$\left| \zeta(3) - \frac{b_n}{a_n} \right| < \frac{C}{a_n^2}, \qquad a_n \sim \frac{(1+\sqrt{2})^{4n}}{n^{3/2}} \cdot \text{const},$$

so the error decays like (√2 − 1)⁴ⁿ ≈ (0.02943…)ⁿ.

3. The clincher: By the Prime Number Theorem, lcm(1,…,n) ≈ eⁿ. If ζ(3) = p/q were rational, then qₙ = 2·lcm(1,…,n)³·aₙ multiplied by |ζ(3) − bₙ/aₙ| would give a nonzero integer of absolute value less than $$C \cdot e^{3n} (\sqrt{2}-1)^{4n} \to 0,$$ since e³·(√2−1)⁴ ≈ 20.09 × 0.0294 ≈ 0.591 < 1. A nonzero integer cannot be smaller than 1 — contradiction. Hence ζ(3) ∉ ℚ. ∎

The critical numerical miracle is that 3 (from lcm³) is just barely small enough: 4 ln(1+√2) ≈ 3.526 > 3.

3.4 Beukers' Elegant Reformulation

Frits Beukers (1979) gave a stunningly short proof via triple integrals:

$$\int_0^1 \!\!\int_0^1 \!\!\int_0^1 \frac{P_n(x) P_n(y)}{1 - (1-xy)z}\, dx\, dy\, dz = A_n + B_n \zeta(3)$$

where Pₙ are shifted Legendre polynomials.

F. Beukers, "A note on the irrationality of ζ(2) and ζ(3)," Bull. London Math. Soc. 11 (1979), 268–272. 📄 https://doi.org/10.1112/blms/11.3.268

4. Known Extensions (Status as of Today)

Still open: Is ζ(3) transcendental? Is ζ(5) irrational? Is ζ(3)/π³ irrational? (Widely believed yes to all; all odd zeta values are conjectured to be algebraically independent of each other and of π.)

Connection to your RH research: ζ(3) appears at s = 3 on the real axis, far from the critical strip, but the Apéry numbers connect to modular forms (Beukers showed a₍ₙ₎ relate to a weight-4 modular form on Γ₀(6)), and the interplay between arithmetic (lcm growth ↔ PNT ↔ zero-free regions of ζ) and analysis in the proof is a beautiful microcosm of zeta-function theory. Also, 1/ζ(3) is the probability that three random integers are coprime.

5. Python Computational Examples

from fractions import Fraction
from math import comb, gcd, isqrt
from functools import reduce
import mpmath as mp

mp.mp.dps = 50  # 50 decimal digits

# ------------------------------------------------------------------
# 1. Reference value
# ------------------------------------------------------------------
zeta3 = mp.zeta(3)
print("zeta(3) =", zeta3)

# ------------------------------------------------------------------
# 2. Naive series vs. Apéry's central-binomial series
# ------------------------------------------------------------------
def zeta3_naive(N):
    return mp.fsum(mp.mpf(1)/mp.mpf(n)**3 for n in range(1, N+1))

def zeta3_apery_series(N):
    """zeta(3) = (5/2) * sum (-1)^(n-1) / (n^3 * C(2n,n))"""
    s = mp.mpf(0)
    for n in range(1, N+1):
        s += mp.mpf((-1)**(n-1)) / (n**3 * comb(2*n, n))
    return mp.mpf(5)/2 * s

print("\nConvergence comparison (error vs true value):")
for N in [5, 10, 20]:
    e1 = abs(zeta3_naive(N) - zeta3)
    e2 = abs(zeta3_apery_series(N) - zeta3)
    print(f"  N={N:3d}   naive err = {mp.nstr(e1,3):>12}   Apéry err = {mp.nstr(e2,3)}")
# The Apéry series gains ~1.2 digits per term; the naive series is hopeless.

# ------------------------------------------------------------------
# 3. The Apéry recurrence:  n^3 u_n = (34n^3-51n^2+27n-5) u_{n-1} - (n-1)^3 u_{n-2}
#    a_n : 1, 5, 73, 1445, ...   (integers -- the miracle)
#    b_n : 0, 6, 351/4, ...      (denominators divide 2*lcm(1..n)^3)
# ------------------------------------------------------------------
def apery_sequences(N):
    a = [Fraction(1), Fraction(5)]
    b = [Fraction(0), Fraction(6)]
    for n in range(2, N+1):
        P = 34*n**3 - 51*n**2 + 27*n - 5
        a.append((P*a[n-1] - (n-1)**3 * a[n-2]) / n**3)
        b.append((P*b[n-1] - (n-1)**3 * b[n-2]) / n**3)
    return a, b

N = 20
a, b = apery_sequences(N)

print("\nApéry numbers a_n (must all be integers!):")
print([int(x) for x in a[:6]])          # 1, 5, 73, 1445, 33001, 819005

print("\nb_n/a_n converging to zeta(3):")
for n in [2, 5, 10, 20]:
    approx = mp.mpf(b[n].numerator) / mp.mpf(b[n].denominator) / \
             (mp.mpf(a[n].numerator) / mp.mpf(a[n].denominator))
    err = abs(approx - zeta3)
    bound = (mp.sqrt(2) - 1)**(4*n)
    print(f"  n={n:2d}  err = {mp.nstr(err,3):>12}   (sqrt2-1)^(4n) = {mp.nstr(bound,3)}")

# ------------------------------------------------------------------
# 4. Verify the closed form  a_n = sum C(n,k)^2 C(n+k,k)^2
# ------------------------------------------------------------------
def apery_closed(n):
    return sum(comb(n,k)**2 * comb(n+k,k)**2 for k in range(n+1))

assert all(apery_closed(n) == int(a[n]) for n in range(N+1))
print("\nClosed form for a_n verified up to n =", N)

# ------------------------------------------------------------------
# 5. Verify integrality of 2*lcm(1..n)^3 * b_n  (the arithmetic miracle)
# ------------------------------------------------------------------
def lcm_upto(n):
    return reduce(lambda x, y: x*y // gcd(x, y), range(1, n+1), 1)

print("\nChecking 2*d_n^3*b_n is an integer (d_n = lcm(1..n)):")
for n in range(1, 13):
    val = 2 * lcm_upto(n)**3 * b[n]
    print(f"  n={n:2d}  integer: {val.denominator == 1}")

# ------------------------------------------------------------------
# 6. The irrationality mechanism: |q_n*zeta(3) - p_n| -> 0
#    with p_n, q_n integers  =>  zeta(3) is irrational
# ------------------------------------------------------------------
print("\nLinear forms |q_n*zeta(3) - p_n| (integers p_n, q_n):")
for n in [4, 8, 12, 16, 20]:
    d3 = 2 * lcm_upto(n)**3
    q  = d3 * int(a[n])
    p  = int(d3 * b[n])
    form = abs(q * zeta3 - p)
    print(f"  n={n:2d}  |q*z3 - p| = {mp.nstr(form, 4)}")
# These tend to 0 geometrically (~0.591^n), yet are nonzero -> contradiction
# if zeta(3) were rational. QED (Apéry, 1979).

Expected highlights of the output:

• a_n = 1, 5, 73, 1445, 33001, 819005, … (OEIS A005259, the Apéry numbers)
• The b/a error at n = 20 is already below 10⁻³⁰, matching (√2−1)⁸⁰
• The linear forms |qζ(3) − p| shrink toward 0 — the visible engine of the proof

6. Key References for Your Paper

1. R. Apéry, Astérisque 61 (1979), 11–13 — http://www.numdam.org/item/AST_1979__61__11_0/
2. A. van der Poorten, Math. Intelligencer 1 (1979), 195–203 — https://doi.org/10.1007/BF03028234
3. F. Beukers, Bull. LMS 11 (1979), 268–272 — https://doi.org/10.1112/blms/11.3.268
4. K. Ball & T. Rivoal, Invent. Math. 146 (2001), 193–207 — https://doi.org/10.1007/s002220100168
5. W. Zudilin, arXiv:math/0104249 — https://arxiv.org/abs/math/0104249
6. Survey: S. Fischler, "Irrationalité de valeurs de zêta," Séminaire Bourbaki 910 (2002/03) — https://arxiv.org/abs/math/0303066
7. OEIS A005259 (Apéry numbers) — https://oeis.org/A005259