ζ(3) is the value of the Riemann zeta function at 3: the sum of 1/n³ over all positive integers. For even inputs, Euler found beautiful closed forms: ζ(2) = π²/6, ζ(4) = π⁴/90, ζ(6) = π⁶/945. For odd inputs, no such formula exists. Whether ζ(3) involves π at all is unknown.
z(3) sits between two values with known closed forms involving pi. Whether z(3) involves pi is still unknown.
In 1978, Roger Apéry announced a proof that ζ(3) is irrational. The audience was sceptical. Henri Cohen and other mathematicians raced home to check it on computers overnight. By the next morning they confirmed it was correct. "It was like thunder in a clear sky," said one attendee. Apéry was 64 years old.
The partial sums 1 + 1/8 + 1/27 + 1/64... approach ζ(3) ≈ 1.20206 from below. Convergence is slow: even at n=50 the sum is still 0.003 away.
Whether ζ(3) can be expressed in terms of π is the outstanding open question. All even zeta values are rational multiples of the corresponding power of π. Odd zeta values seem to live in a different world. Infinitely many odd values ζ(2n+1) are known to be irrational (Rivoal, 2000), but the exact pattern remains mysterious. Full value: 1.20205690315959428539973816151144999…
ζ(2k) = rational number × π^(2k) for all even k. Euler proved this for all even values. But ζ(3), ζ(5), ζ(7)... are completely different. ζ(3) is irrational (Apéry), but no relation to π is known. It may be truly independent of π.
Table showing zeta at even integers known as pi fractions but odd integers unknown
| Even s: exact formulas | Odd s: mystery |
|---|---|
| ζ(2) = π²/6 | ζ(3) = 1.20206... |
| ζ(4) = π⁴/90 | irrational (Apéry 1978) |
| ζ(6) = π⁶/945 | ζ(5) = 1.03693... |
| ζ(8) = π⁸/9450 | irrational? unknown |
| All = rational × π^s | No π connection known |
Unknown. Roger Apery proved in 1978 that zeta(3) is irrational, but whether it is transcendental remains an open problem. It is widely believed to be transcendental, but no proof exists.
In quantum electrodynamics (corrections to the electron magnetic moment), random matrix theory, and the entropy of a two-dimensional Ising model. It appears in the Fermi-Dirac and Bose-Einstein distributions in statistical mechanics.
Ramanujan found rapidly converging series for zeta(3), including a formula involving 7pi^3/180 and sums over exponentials. His notebooks contained dozens of identities related to zeta(3), most proved only decades after his death.
Integers A(n) = sum of C(n,k)^2 C(n+k,k)^2 over k, which appear in Apery's proof of irrationality. The first few are 1, 5, 73, 1445, 33001. They satisfy a recurrence relation and grow in a way that forces the denominators of partial sums of 1/n^3 to cancel specific factors, making the limit irrational.
Apery's constant zeta(3) is the sum 1 + 1/8 + 1/27 + 1/64 + ... = 1.20205690315959. For even values of s, Euler found closed forms involving pi: zeta(2) = pi^2/6, zeta(4) = pi^4/90. For odd values no such formula is known. Roger Apery proved zeta(3) is irrational in 1978 at age 64. Whether it is transcendental, or expressible in terms of pi, remains unknown.