Weil’s explicit formula · positivity criterion

One number, counted two ways

Pick a test function. The zeros of ζ give it a value. The primes give it the same value. The Riemann Hypothesis is the claim that this value is never negative.

σ — concentration0.60

f has width ≈ 1.20 in u, ≈ 1.67 in r

r₀ — test frequency14.13

f oscillates as cos(r₀u); primes enter as cos(r₀ log n)

Aim at

Arithmetic side

g(u) samples the prime powers

g = f ⋆ f̃ · weight Λ(n)/√n at u = log n · 15 in view

Brass stems are Λ(n)/√n. Dots sit on g. Blue dots push W up, oxblood pushes it down — the prime term enters with a minus sign, so a positive g at a prime subtracts.

Analytic side

h(r) samples the zeros

h = |f̂|² · ticks at γ · 1 of 5 in view carry weight

Every tick sits where h ≥ 0, because on the real axis h is a modulus squared. Narrow the window (raise σ) and h resolves individual zeros; widen it and it sees many at once — but then g stops reaching past the small primes.

The ledger

W(f) — assembled from the primes, checked against the zeros

σ = 0.60 · r₀ = 14.13

h(i/2) + h(−i/2)

the pole of ζ at s = 1

8.23e-31

− g(0) log π

conductor

-2.4348

(1/2π) ∫ h(r) Re ψ(¼ + ir/2) dr

archimedean place

4.1506

− 2 Σ Λ(n) n^(−½) g(log n)

finite places — the primes

2.8081

W(f) = Σ_ρ h(γ_ρ)

4.5239

independently: 2 Σ_γ>0 h(γ), first 50 zeros

the two sides close to 5e-10 of the term scale — this is the explicit formula

4.5239

Where the sign can fail

Re h(z) across the strip |Im z| ≤ ½

blue: Re h > 0 · oxblood: Re h < 0 · shaded by phase, Re h / |h|

Im z = 0 — h = |f̂|² ≥ 0 here, always

τ = 14.13

r₀

Zeros come in quadruples ρ, 1−ρ, ρ̄, 1−ρ̄, so an off-line zero puts four markers at (±τ, ±β). Their total contribution to W is 4·Re h(τ + iβ). Slide β and watch the markers walk off the white line into whatever lobe they land in.

quadruple contributes

+4.524

on the line — a pair, not a quadruple

τ — which zero goes rogue

β — distance off the critical line0.000

β = 0 — RH holds, nothing to see

raise β first

Raise β above zero to see how fast an off-line zero outgrows everything sitting on the line.

How to read this

The easy direction. On the real axis h = |f̂|², so every real zero contributes a nonnegative amount. If all γ are real, W ≥ 0 term by term. That is the whole of it.

The hard direction. Off the axis, h is only an analytic continuation and carries a phase. The sign map shows lobes where Re h < 0. A zero landing in one of those lobes drags W below zero, and Weil’s converse says you can always arrange that.

Why this is not a numerical experiment. The rogue quadruple only outweighs the archimedean terms past σ* = √(π/2)/β, and by then g(log n) has barely decayed, so the prime sum needs n far beyond anything summable. The criterion is sharp and inaccessible at once.

Prime sum truncated at n ≤ 10⁶; zero sum at the first 50 γ. Both are converged to ~10⁻¹⁰ over the σ range allowed here. Identity verified against mpmath to 9 digits.