Convergence — Watch Both Sides Approach the Same Value
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Sum Σ 1/ns (naturals)
Product Π 1/(1−p−s) (primes)
Difference
Side by Side — Partial Sums & Products
Prime Product Expansion
Each prime p contributes a factor 1/(1−p−s). When expanded as geometric series and multiplied together, they generate all naturals exactly once.
Current Values
Sum (naturals)
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Product (primes)
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Difference
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The Discovery
Euler, 1737
The left side sums over all natural numbers: 1, 2, 3, 4, 5, 6, 7...
The right side multiplies over only primes: 2, 3, 5, 7, 11, 13...
They produce exactly the same value.
This identity reveals that primes are the atoms of arithmetic — every natural number is uniquely built from prime factors.
The left side sums over all natural numbers: 1, 2, 3, 4, 5, 6, 7...
The right side multiplies over only primes: 2, 3, 5, 7, 11, 13...
They produce exactly the same value.
This identity reveals that primes are the atoms of arithmetic — every natural number is uniquely built from prime factors.
Special Values
s = 2 (Basel Problem)
ζ(2) = π²/6 ≈ 1.6449
Euler's breakthrough that stunned Europe
s = 3 (Apéry's Constant)
ζ(3) ≈ 1.2020
Proven irrational in 1978
s = 4
ζ(4) = π⁴/90 ≈ 1.0823
Related to π via Bernoulli numbers
ζ(2) = π²/6 ≈ 1.6449
Euler's breakthrough that stunned Europe
s = 3 (Apéry's Constant)
ζ(3) ≈ 1.2020
Proven irrational in 1978
s = 4
ζ(4) = π⁴/90 ≈ 1.0823
Related to π via Bernoulli numbers