# Basel Problem

**Type:** Theorem  
**Domain:** Named Results / Analysis  
**Posed:** 1650, Pietro Mengoli  
**Proved:** 1734, Leonhard Euler  
**Codex URL:** /mathematics/named-results/basel-problem/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Sum of reciprocal squares: 1+1/4+1/9+1/16+... = π²/6. Connects π to number theory.

## Euler's heuristic proof (1734)
Treated sin(x)/x as an "infinite polynomial," factored using roots at ±π,±2π,...; comparing x² coefficients between Taylor series and product form gives π²/6.

## Rigorous follow-up
Euler (1741) gave a fully rigorous proof; modern proofs exist via double integration, Fourier series, complex analysis (residue theorem).

## Zeta function connection
Basel Problem = ζ(2) = π²/6. Euler found general formula for ζ(2n) for all positive integers n (involving Bernoulli numbers). Odd values (ζ(3), ζ(5)...) remain largely mysterious — Apéry (1978) proved ζ(3) irrational ("Apéry's constant"); no closed form known.

## Application: coprime probability
Probability two random positive integers are coprime = 1/ζ(2) = 6/π² ≈ 60.79%.

## Sources
### Tier 1
- Euler, L. (1741). De summis serierum reciprocarum.
- Ayoub, R. (1974). Euler and the zeta function. *Amer. Math. Monthly*.
### Tier 2
- van der Poorten, A. (1979). *Math. Intelligencer*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*