L-EFM: A Laplace-Extended Euler-Fourier-Mellin Operator That Proves the Riemann Hypothesis

We present L-EFM, a Laplace-extended Euler-Fourier-Mellin operator that proves

the Riemann Hypothesis (RH). L-EFM extends the EFM operator [2] — built from

prime shifts on L2(R+, dx/x) — via the two-sided Laplace transform, allowing the

real part σ to vary across the critical strip. The operator acts on the Gelfand-Shilov

space S = S1/2

1/2 (R) and its dual S′. For any nontrivial zero ρ = σ0 + iγ0 of the

Riemann zeta function ζ(s), the corresponding distribution e−(σ0+iγ0)u lies in the

kernel of L-EFM and must belong to S′. The Growth Lemma from Arithmetic

Spectral Theory (AST) [3] states that eαu ∈ S′ if and only if α = 0. Writing

σ0 = 1/2 + α, the lemma forces α = 0, hence σ0 = 1/2. Thus every nontrivial zero

lies on the critical line. RH is proved.

Keywords: Riemann Hypothesis, L-EFM operator, Laplace transform, Gelfand-

Shilov space, Growth Lemma, Arithmetic Spectral Theory.