Euler's product ζ(s) = ∏ₚ (1 − p^{−s})⁻¹ builds the zeta function from the primes — but only where there are no zeros. Where it converges it proves the zeros away; where the zeros live it never settles. Three live panels, all computed here from a prime sieve to one million: where the product converges, where it swings, and the compensated series whose convergence to the right of the line is the Riemann Hypothesis itself.
Instruments, not proofs. Every curve is a finite window. Finite windows can refute and never prove. Nothing on this page bears on the truth of the Riemann Hypothesis.
computing — sieving to one million…
For Re s > 1 the product converges absolutely and is never zero: a convergent product of nonzero factors is nonzero. That is the product's one theorem about zeros — there are none where it converges — and it proves it completely. Here the finite products over primes ≤ X close on ζ(2) = π²/6 like clockwork.
At a zero ρ = ½ + iγ the true value is ζ(ρ) = 0 — computed independently here from the alternating series, not from any table. The finite products over primes ≤ X do not approach it: they never converge and never stop. Their logarithm carries the prime sum's rotating main term Re[X^{1−s}/((1−s)·log X)], of size √X/(|1−s|·log X) (dashed): the swing follows its sign, with a slowly varying offset from the zero-side terms. The same happens at a height that is not a zero (the t = 15 control, where |ζ| = 0.72): the swing is about the strip, not about the zero. No finite product has a zero; the infinite one does not exist here as a product.
Remove the pole and the product becomes an honest Dirichlet series: Σ (Λ(n) − 1)·n^{−s} — the primes minus the integers, where Λ(n) = log p if n is a power of the prime p and 0 otherwise. Its partial sums settle to the right of the line (s = 0.9, 0.75, 0.6) and wander at and left of it (s = 0.5, 0.4). This picture is a finite window: it illustrates the theorem below, it does not test it.
Theorem (classical). The abscissa of convergence of Σ (Λ(n) − 1)·n^{−s} is exactly Θ, the supremum of the real parts of the zeros of ζ. Hence the Riemann Hypothesis holds if and only if this series converges for every s with Re s > ½.
Why (⟹): under the hypothesis, ψ(x) − x = O(√x·log² x) (von Koch), and Σ_{n≤x}(Λ(n) − 1) = ψ(x) − x + O(1); partial summation then gives convergence for Re s > ½.
Why (⟸): a Dirichlet series that converges at one point converges, and is analytic, in the open half-plane to its right. This series equals −ζ′(s)/ζ(s) − ζ(s) for Re s > 1, so convergence on Re s > ½ continues −ζ′/ζ analytically there: no zeros with Re s > ½ — and none with Re s < ½, by the functional equation's mirror.
So the edge of the product's convergence is the rightmost zero. "Build the Euler product over all primes to the right of the line" is not a step toward the Riemann Hypothesis — it is the Riemann Hypothesis. Same species as "boundedness of the truncated Weil form alone is RH" (von Koch): the circularity lint at maximum precision.
Every number on this page is recomputed in your browser from the sieve and compared with the committed Python twin (receipts/product_edge_receipts.py). The ten zero heights are Odlyzko's, and each is verified here by |ζ(½ + iγ)| ≈ 0 from the alternating series — a control height that is not a zero must fail that test.
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