Equation 43: Euler's Formula for ζ(2k) — In-Depth Analysis

$$\zeta(2k) = \frac{(-1)^{k+1}(2\pi)^{2k}B_{2k}}{2(2k)!}, \qquad k = 1, 2, 3, \dots$$

1. Historical Background

The Basel Problem (1650–1735)

The story begins with the Basel Problem, first posed by Pietro Mengoli in 1650: find the exact value of

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$$

The problem defeated the greatest mathematicians of the era, including Jakob Bernoulli, Johann Bernoulli, Leibniz, and John Wallis (who computed it numerically to three decimal places). It became famous enough that solving it guaranteed instant mathematical celebrity. The name "Basel Problem" honors the city of Basel, Switzerland — home of the Bernoullis and of the man who solved it.

Euler's Triumph (1734–1740)

Leonhard Euler (1707–1783) announced his solution in 1734/1735, published in "De summis serierum reciprocarum" (On the sums of series of reciprocals), Commentarii academiae scientiarum Petropolitanae 7 (1740), pp. 123–134 — catalogued as E41 in the Eneström index:

$$\zeta(2) = \frac{\pi^2}{6}$$

Euler's original argument was audacious: he treated $\sin(\pi z)/(\pi z)$ as an "infinite polynomial" and factored it over its roots — exactly what your Method A formalizes. The infinite product

$$\frac{\sin \pi z}{\pi z} = \prod_{n=1}^{\infty}\left(1 - \frac{z^2}{n^2}\right)$$

was not rigorously justified until Weierstrass's factorization theorem (1876), over a century later. Euler, aware of the gap, verified his result numerically using what we now call the Euler–Maclaurin formula and later gave additional proofs.

Euler didn't stop at $\zeta(2)$. In the same paper and in later work (notably E130, "De seriebus quibusdam considerationes", 1750, and his Institutiones calculi differentialis, 1755), he derived the general formula — Equation 43 — connecting $\zeta(2k)$ to the Bernoulli numbers $B_{2k}$, which Jakob Bernoulli had introduced posthumously in Ars Conjectandi (1713) in the context of sums of powers.

First few values:

Connection to Riemann (1859)

Bernhard Riemann, in his revolutionary 8-page paper "Über die Anzahl der Primzahlen unter einer gegebenen Größe" (On the Number of Primes Less Than a Given Magnitude, 1859), extended $\zeta(s)$ to a meromorphic function on all of $\mathbb{C}$ and proved the functional equation:

$$\zeta(s) = 2^s \pi^{s-1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma(1-s)\, \zeta(1-s)$$

Equation 43 is deeply tied to this: applying the functional equation to $\zeta(2k)$ yields the values at negative odd integers,

$$\zeta(1-2k) = -\frac{B_{2k}}{2k},$$

and the trivial zeros $\zeta(-2k) = 0$ arise because $\sin(\pi s/2)$ vanishes at even negative integers. So Euler's even-values formula is, in modern language, one "half" of the functional equation — remarkably, Euler himself conjectured the functional equation in 1749 (E352) based on these values, 110 years before Riemann proved it.

The Odd Values — Still Open

A striking contrast: no closed form like Equation 43 is known for $\zeta(3), \zeta(5), \dots$. Key milestones:

• Apéry (1979): proved $\zeta(3)$ is irrational.
• Rivoal (2000), Ball–Rivoal (2001): infinitely many $\zeta(2k+1)$ are irrational.
• Zudilin (2001): at least one of $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational.

2. Key References and Papers

1. Euler, L. De summis serierum reciprocarum (E41), 1740. Euler Archive: https://scholarlycommons.pacific.edu/euler-works/41/

2. Riemann, B. Über die Anzahl der Primzahlen unter einer gegebenen Größe, Monatsberichte der Berliner Akademie, 1859. Clay Mathematics Institute (original manuscript + translation): https://www.claymath.org/collection/riemanns-1859-manuscript/

3. Ayoub, R. "Euler and the Zeta Function," American Mathematical Monthly 81 (1974), 1067–1086. DOI: https://doi.org/10.2307/2319041 — excellent history of Equation 43.

4. Chapman, R. "Evaluating ζ(2)" — a survey of 14+ distinct proofs of the Basel problem (including your Method B, Parseval): https://empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

5. van der Poorten, A. "A proof that Euler missed... Apéry's proof of the irrationality of ζ(3)," Mathematical Intelligencer 1 (1979), 195–203. DOI: https://doi.org/10.1007/BF03028234

6. Dunham, W. Euler: The Master of Us All, MAA, 1999 — Chapter on the Basel problem.

7. Edwards, H. M. Riemann's Zeta Function, Academic Press, 1974 (Dover reprint) — the standard reference for your RH paper.

(Note: links were correct as of my training data; please verify DOIs, which are the most stable.)

3. Why the Formula Matters for the Riemann Hypothesis

• It shows $\zeta(2k)$ is always a rational multiple of $\pi^{2k}$, hence transcendental (by Lindemann, 1882).
• Via the functional equation it locates the trivial zeros at $s = -2, -4, -6, \dots$, sharpening the RH statement: all non-trivial zeros lie on $\text{Re}(s) = 1/2$.
• The values $\zeta(1-2k) = -B_{2k}/2k$ appear throughout modern number theory (p-adic L-functions, the Kummer congruences, and the von Staudt–Clausen theorem).
• Asymptotically, since $\zeta(2k) \to 1$, Equation 43 gives the growth rate of Bernoulli numbers: $$|B_{2k}| \sim \frac{2(2k)!}{(2\pi)^{2k}} \sim 4\sqrt{\pi k}\left(\frac{k}{\pi e}\right)^{2k}$$

4. Computational Examples in Python

Example 1: Exact symbolic verification (SymPy)

from sympy import bernoulli, zeta, pi, factorial, Rational, simplify

print(f"{'k':>3} | {'B_2k':>12} | {'zeta(2k) closed form':>25} | match")
print("-" * 65)

for k in range(1, 8):
    B2k = bernoulli(2 * k)
    # Equation 43
    closed_form = (-1)**(k + 1) * (2 * pi)**(2 * k) * B2k / (2 * factorial(2 * k))
    sympy_value = zeta(2 * k)
    match = simplify(closed_form - sympy_value) == 0
    print(f"{k:>3} | {str(B2k):>12} | {str(closed_form):>25} | {match}")

Output:

  k |         B_2k |      zeta(2k) closed form | match
-----------------------------------------------------------------
  1 |          1/6 |                   pi**2/6 | True
  2 |        -1/30 |                  pi**4/90 | True
  3 |         1/42 |                 pi**6/945 | True
  4 |        -1/30 |                pi**8/9450 | True
  5 |         5/66 |              pi**10/93555 | True
  6 |   -691/2730 |    691*pi**12/638512875 | True
  7 |          7/6 |       2*pi**14/18243225 | True

Example 2: Bernoulli numbers from scratch + high-precision check (mpmath)

from fractions import Fraction
from math import comb
from mpmath import mp, mpf, pi, factorial, zeta

mp.dps = 50  # 50 decimal digits

def bernoulli_numbers(n_max):
    """Compute B_0..B_n exactly via the recurrence
       sum_{j=0}^{m} C(m+1, j) B_j = 0  (m >= 1), B_0 = 1."""
    B = [Fraction(1)]
    for m in range(1, n_max + 1):
        s = sum(Fraction(comb(m + 1, j)) * B[j] for j in range(m))
        B.append(-s / (m + 1))
    return B

B = bernoulli_numbers(20)

print(f"{'k':>3} | {'Equation 43':>52} | {'error vs mpmath.zeta':>12}")
for k in range(1, 11):
    B2k = mpf(B[2*k].numerator) / mpf(B[2*k].denominator)
    val = (-1)**(k + 1) * (2 * pi)**(2 * k) * B2k / (2 * factorial(2 * k))
    err = abs(val - zeta(2 * k))
    print(f"{k:>3} | {mp.nstr(val, 45):>52} | {mp.nstr(err, 3):>12}")

Errors come out at the level of $10^{-49}$, i.e., pure floating-point round-off — the identity is exact.

Example 3: Convergence of the raw series vs. the closed form

from mpmath import mp, mpf, pi
mp.dps = 30

def zeta_partial(s, N):
    return sum(mpf(1) / mpf(n)**s for n in range(1, N + 1))

exact = pi**2 / 6
print("Basel problem: zeta(2) = pi^2/6 =", mp.nstr(exact, 25))
print(f"\n{'N':>10} | {'partial sum':>28} | {'error':>10}")
for N in [10, 100, 1000, 10_000, 100_000]:
    ps = zeta_partial(2, N)
    print(f"{N:>10} | {mp.nstr(ps, 25):>28} | {mp.nstr(exact - ps, 4):>10}")

Output — note the error is $\approx 1/N$ (this slow convergence is precisely why Wallis's numerical attack failed and Euler needed Euler–Maclaurin acceleration):

         N |                  partial sum |      error
        10 |    1.549767731166540690082006 |  0.09516
       100 |    1.634983900184892261717733 | 0.009950
      1000 |    1.643934566681561458274221 | 0.0009995
     10000 |    1.644834071848064577886583 | 0.00009999
    100000 |    1.644924066898242637633157 | 0.00001000

Example 4: Method B — Parseval / Fourier verification for ζ(2)

import numpy as np

# f(x) = x on (-pi, pi): Fourier coefficients b_n = 2(-1)^{n+1}/n
# Parseval: (1/pi) * integral of x^2 dx = sum b_n^2  =>  2*pi^2/3 = 4*zeta(2)

N = 1_000_000
n = np.arange(1, N + 1)
bn_sq_sum = np.sum((2.0 / n)**2)          # signs vanish when squared
lhs = 2 * np.pi**2 / 3

print("sum b_n^2      =", bn_sq_sum)
print("2*pi^2/3       =", lhs)
print("=> zeta(2)     =", bn_sq_sum / 4)
print("   pi^2/6      =", np.pi**2 / 6)

Example 5: Asymptotics — ζ(2k) → 1 and Bernoulli growth

from mpmath import mp, mpf, pi, factorial, sqrt, e, bernoulli
mp.dps = 30

print(f"{'k':>3} | {'zeta(2k) - 1':>14} | {'|B_2k|':>16} | {'Stirling estimate':>18}")
for k in [1, 2, 5, 10, 20, 30]:
    z = (-1)**(k+1) * (2*pi)**(2*k) * bernoulli(2*k) / (2*factorial(2*k))
    est = 4 * sqrt(pi * k) * (k / (pi * e))**(2 * k)   # asymptotic |B_2k|
    print(f"{k:>3} | {mp.nstr(z - 1, 6):>14} | {mp.nstr(abs(bernoulli(2*k)), 8):>16} | {mp.nstr(est, 8):>18}")

This demonstrates $\zeta(2k) - 1 \approx 2^{-2k}$ and confirms the factorial explosion of $|B_{2k}|$ — the reason the zeta function's "even values" encode such deep arithmetic.

5. Suggested Framing for Your Paper

1. Narrative arc: Mengoli's problem → Euler's non-rigorous product (Method A) → rigorous 19th-century proofs (Parseval/Method B, Weierstrass) → Riemann's functional equation reinterpreting Euler's formula → the open problem of odd values → RH itself.
2. Key point to emphasize: Equation 43 + the functional equation together explain all the trivial zeros — RH is a statement about everything that remains.
3. Cite Ayoub (1974) for history, Edwards (1974) for the analytic machinery, and Chapman's survey for the zoo of proofs of the $k=1$ case.