Work information

Title A Complete Proof of the Riemann Hypothesis via Variational Spectral Theory
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
Work type Education, Informative
Tags the riemann hypothesis in full

-------------------------

Registry info in Safe Creative

Identifier 2509012957367
Entry date Sep 1, 2025, 11:13 PM UTC
License All rights reserved

-------------------------

Copyright registered declarations

Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Sep 1, 2025.


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