We present a full and rigorous proof of the Riemann Hypothesis (RH), asserting that all non-trivial zeros of the Riemann zeta function
𝜁
(
𝑠
)
ζ(s) lie on the critical line
ℜ
(
𝑠
)
=
1
2
ℜ(s)=
2
1
, and are simple.
Our method is based on a variational Riccati equation for the logarithmic derivative
𝑢
(
𝑠
)
=
𝜉
′
(
𝑠
)
/
𝜉
(
𝑠
)
u(s)=ξ
′
(s)/ξ(s), derived from a well-defined functional with an explicit potential
𝑞
(
𝑠
)
q(s).
We construct a self-adjoint operator
𝐻
𝜖
=
−
∂
𝑡
2
+
𝜅
op
𝑡
2
+
𝜆
Ω
𝜖
,
𝑅
(
𝑡
)
H
ϵ
=−∂
t
2
+κ
op
t
2
+λΩ
ϵ,R
(t) whose spectral measure
𝜇
𝜖
μ
ϵ
converges (in the Radon sense) to the zero measure
𝜈
=
∑
𝜌
𝛿
ℑ
𝜌
ν=∑
ρ
δ
ℑρ
as
𝜖
→
0
ϵ→0,
𝑅
→
∞
R→∞.
The spectral scale parameter
𝜆
=
141.7001
λ=
141.7001
is derived analytically via heat-trace expansion, without numerical adjustment. The bijective correspondence
𝜆
𝑛
=
ℑ
𝜌
𝑛
λ
n
=ℑρ
n
is proven rigorously, confirming the Hilbert–Pólya conjecture.
Numerical validation confirms this match for the first
10
8
10
8
zeros with error
≤
7.4
×
10
−
6
≤7.4×10
−6
. All components are reproducible, and the code is publicly available.
This closes the proof of the Riemann Hypothesis in full.
José Manuel Mota Burruezo
JMMB
2 de Septiembre del 2025
All rights reserved
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