About the work
We propose a proof of the Riemann Hypothesis (RH). Starting from a variational principle, we derive a Riccati equation for
𝑢
(
𝑠
)
=
𝜉
′
(
𝑠
)
/
𝜉
(
𝑠
)
u(s)=ξ
′
(s)/ξ(s) with an explicit meromorphic
𝑞
(
𝑠
)
q(s), and prove global uniqueness via Nevanlinna and Phragmén–Lindelöf under symmetry, growth, and pole structure. We construct a self-adjoint operator
𝐻
𝜀
H
ε
(Friedrichs extension; Kato–Rellich/KLMN), and establish spectral–zero measure equality in the Radon sense with explicit error bounds
∣
𝐴
𝜀
[
𝜑
]
∣
≤
𝜁
(
2
)
𝜀
1
−
𝜀
∥
𝜑
∥
𝐶
2
+
𝜋
𝑅
∥
𝜑
∥
𝐶
1
∣A
ε
[φ]∣≤ζ(2)
1−ε
ε
∥φ∥
C
2
+
R
π
∥φ∥
C
1
. Together with a “no-ghosts/no-gaps” lemma for purely atomic Radon measures, this yields an exact one-to-one correspondence between the spectrum and the imaginary parts of the nontrivial zeros. Zeros off the critical line are excluded by a contour argument using the functional symmetry
𝜉
(
1
−
𝑠
)
=
𝜉
(
𝑠
)
ξ(1−s)=ξ(s) and Stirling estimates. Simplicity of zeros follows conditionally from 1D Sturm–Liouville spectral simplicity and, independently, from the local Riccati expansion. Numerical experiments (up to
10
5
10
5
zeros; interval arithmetic) provide supporting evidence but are not part of the proof. The work is presented as a proposed proof, subject to rigorous peer review.
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Title Variational Riccati Framework and Spectral–Zero Measure Equality for the Riemann Hypothesis
We propose a proof of the Riemann Hypothesis (RH). Starting from a variational principle, we derive a Riccati equation for
𝑢
(
𝑠
)
=
𝜉
′
(
𝑠
)
/
𝜉
(
𝑠
)
u(s)=ξ
′
(s)/ξ(s) with an explicit meromorphic
𝑞
(
𝑠
)
q(s), and prove global uniqueness via Nevanlinna and Phragmén–Lindelöf under symmetry, growth, and pole structure. We construct a self-adjoint operator
𝐻
𝜀
H
ε
(Friedrichs extension; Kato–Rellich/KLMN), and establish spectral–zero measure equality in the Radon sense with explicit error bounds
∣
𝐴
𝜀
[
𝜑
]
∣
≤
𝜁
(
2
)
𝜀
1
−
𝜀
∥
𝜑
∥
𝐶
2
+
𝜋
𝑅
∥
𝜑
∥
𝐶
1
∣A
ε
[φ]∣≤ζ(2)
1−ε
ε
∥φ∥
C
2
+
R
π
∥φ∥
C
1
. Together with a “no-ghosts/no-gaps” lemma for purely atomic Radon measures, this yields an exact one-to-one correspondence between the spectrum and the imaginary parts of the nontrivial zeros. Zeros off the critical line are excluded by a contour argument using the functional symmetry
𝜉
(
1
−
𝑠
)
=
𝜉
(
𝑠
)
ξ(1−s)=ξ(s) and Stirling estimates. Simplicity of zeros follows conditionally from 1D Sturm–Liouville spectral simplicity and, independently, from the local Riccati expansion. Numerical experiments (up to
10
5
10
5
zeros; interval arithmetic) provide supporting evidence but are not part of the proof. The work is presented as a proposed proof, subject to rigorous peer review.
Work type Education, Informative
Tags the riemann hypothesis
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Registry info in Safe Creative
Identifier 2508282922464
Entry date Aug 28, 2025, 2:17 PM UTC
License All rights reserved
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Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Aug 28, 2025.
Information available at https://www.safecreative.org/work/2508282922464-variational-riccati-framework-and-spectral-zero-measure-equality-for-the-riemann-hypothesis
/
Registration: 2508282922464
