Each zero ½ + iγ of the Riemann zeta function becomes a point with angles θp = γ log p mod 2π. The critical line turns into a spiral of irrational slope that never closes, and the zeros are marked points along it. The surface tint is Landau’s prediction: zeros avoid angle 0 for every prime. Further down, each panel adds a layer to the same torus, ending with what a finite window of Weil’s quadratic form can certify.
The functional equation pairs ρ with 1 − ρ̄. On the torus that is the mirror r ↦ 1/r, and the surface is its fixed set: the seam. An off-line pair sits at tube radius q±δ, one point on each side.
| x | Landau | zeros |
|---|
Landau’s formula, which holds without assuming RH, gives −A·log p / √(pk) for x = pk and 0 for anything else. A = (T₂ − T₁)/2πN ≈ ; random scatter would be about 1/√N ≈ 0.015.
Every panel on this page is a layer on the same torus. Each button sets the 3-D view to show that layer.
The arithmetic lives in the angles. For every prime p, zeros avoid θp = 0 and gather near π. Landau’s formula predicts this without assuming RH, and the surface tint shows it.
The Euler product is a function on the torus. Its truncation is smallest where every angle is π, and the zeros sit where it is small.
Leaving the critical line means leaving the surface. An off-line pair sits at tube radius q±δ, one point on each side of the seam. It changes Weil’s quadratic form by only about δ².
Where such a pair can hide depends on the angles. Dense zeros near θ ≈ π hide it best, so the hiding map follows Landau’s tint.
A window sees a fixed part of this picture. A test function supported in [−a, a] reads prime powers up to x = e2a, and its certified upper bound on λ charges each zero by height alone. The charge is the same at every angle and the same on or off the surface.
A different L-function has a different torus. Borromean primes are pairwise unlinked from p₁ and p₂ but linked as a triple. On the torus of the Rédei L-function, reached by three quadratic steps, they invert Landau’s bias.
Layer 5 explains why the upper bound also holds for the Davenport–Heilbronn function. It has a functional equation and a zero count of the same form and satisfies the same kind of bound, yet it has zeros off the line. A positive lower bound for every window would imply RH, so it has to use something Davenport–Heilbronn lacks. The natural candidate is a tempered Euler product, one whose local factors obey the Ramanujan bound. An Euler product alone is not enough: the octonion lattice’s zeta, ζ(s)ζ(s−3), has one, yet none of its zeros lie on its center line. On this torus, ζ’s Euler product appears as the angle structure of layers 1, 2 and 4.
The same torus cut open into a square. The spiral is a straight line of slope log q / log p that wraps at the edges. Shading is the predicted density.
Bars: all 4,491 zeros in 24 bins, relative to a uniform spread (1.0). Curve: Landau’s prediction. Each prime’s angles are pushed away from 0.
Weil’s criterion turns RH into positivity of a quadratic form. Restrict the test functions to a window of length log x and the form has a smallest eigenvalue λ, which falls very fast as the window grows. An off-line pair like the one in the 3-D view changes the form by only about δ², so the size of λ is what a window can resolve. For ζ and for L(s, χ) of conductor q the natural scale is c = 2πx/q.
| Statement | Status | Note |
|---|---|---|
| Every zero, on or off the line, is a zero of the test functions (Theorem 1), which gives an upper bound on λ from zero counts alone (Theorem 2) | proved | Two corrections from the 3 Oct review are pending: a narrower Mellin strip and the range of the zero count. Neither changes a number. |
| The bounds in the left chart, at every point shown (Theorem C) | certified | Interval arithmetic in Arb. Finite statements only. |
| Hermite rate: λ ≤ K·cd+1e−c for all x ≥ 2q (Proposition H) | proved | d is the degree of the trial’s polynomial. The constants K are certified. |
| Polynomial bounds on the out-of-band terms (M2) | proved | Derived and audited on 3 Oct; coarse constants. |
| A lower bound ‖fwin‖² ≥ e2βc−p/C for the window norm (M1) | open | The middle chart shows power-law decay with no exponential loss. The missing piece is an explicit remainder after the cancellation. |
| Kaiser–Bessel rate e−2c for all x (Conjecture R) | open | Follows from M1 and M2. |
| A matching lower bound on λ | open | As hard as RH: a positive lower bound for every window would imply it. |
Data: this work’s R2 theorem sheet (Theorem C, certified with Arb) and the audit of 3 Oct 2026. Degree 1 only.
Three primes can be pairwise unlinked yet linked as a triple, like the Borromean rings. For primes p₁, p₂ ≡ 1 mod 4 with (p₁/p₂) = 1, Rédei’s symbol [p₁, p₂, p] = ±1 is the arithmetic version of Milnor’s triple linking number (Morishita). It lives in a degree-8 field R with Galois group D₄ and in that field’s 2-dimensional L-function L(s, ρ). The coefficient of L(s, ρ) at p is 0 when p is linked to p₁ or p₂. When p is not linked to either, the coefficient is 2·[p₁, p₂, p].
Landau’s formula reads that coefficient from the zeros. On the torus of L(s, ρ), the zeros avoid θp = 0 when the coefficient is +2 and gather there when it is −2. The chart reads it for every prime below 1000, from the zeros of L(s, ρ) for (p₁, p₂) = (5, 29) up to height 1000.
R(p) = −S(p) / [(T/4π) log p/√p], with S(p) a Fejér-weighted sum of cos(γ log p) over the zeros. Every unlinked prime reads its coefficient to within . On the zeros of L(χ₅)L(χ₂₉), which has the same conductor and gamma factor but no triple linking, the same primes read +2.
| p₃ | [5, 29, p₃] | read from L(ρ) | read from L(χ₅)L(χ₂₉) |
|---|
The explicit formula decides this readout in advance, so it is a check of the zeros and coefficients, not a new result (step 1 in docs/BORROMEAN.md). The 2087 zeros are complete to height 1000 by Turing’s method. The open question is whether the window minimum of L(s, ρ) differs from that of L(χ₅)L(χ₂₉), which has the same invariants; that is step 2.
Zeros in (100, 5000] from rh-zero-torus (this work), computed with the Riemann–Siegel formula. Count: 4,491 (Riemann–von Mangoldt ≈ 4491.3).