Work information

Title A Complete Proof of the Riemann Hypothesis via S-Finite Adelic Systems (DR) by José Manuel Mota B.
Abstract:
We construct an entire function
𝐷
(
𝑠
)
D(s) of order ≤ 1 satisfying
𝐷
(
1
−
𝑠
)
=
𝐷
(
𝑠
)
D(1−s)=D(s) and

lim
⁡
𝜎
→
+
∞
log
⁡
𝐷
(
𝜎
+
𝑖
𝑡
)
=
0
,
σ→+∞
lim
​

logD(σ+it)=0,

via S-finite adelic smoothing and relative Fredholm determinants. No reference to the Riemann zeta function
𝜁
(
𝑠
)
ζ(s) or the completed function
Ξ
(
𝑠
)
Ξ(s) is made in Sections 1–2. Using Schatten-class bounds and double operator integrals, we rigorously justify all limit exchanges and derive a complete explicit formula for
(
log
⁡
𝐷
)
′
(logD)
′
, including the exact Archimedean term and residues at
𝑠
=
0
,
1
s=0,1.

A holomorphic self-adjoint ratio determinant
Dratio
(
𝑠
)
:
=
det
⁡
(
(
𝐴
𝑆
,
𝛿
−
𝑠
)
(
𝐴
0
−
𝑠
)
−
1
)
Dratio(s):=det((A
S,δ
​

−s)(A
0
​

−s)
−1
) is shown to be non-vanishing off the critical line
ℜ
𝑠
=
1
2
ℜs=
2
1
​

, and identified with
𝐷
(
𝑠
)
D(s) via two-line Paley–Wiener uniqueness. Matching explicit formulas on both vertical lines and normalization at
∞
∞ yield

𝐷
≡
Ξ
.
D≡Ξ.

This implies that all non-trivial zeros of
𝜁
(
𝑠
)
ζ(s) lie on the critical line.

Key achievements:

Entire function
𝐷
(
𝑠
)
D(s) constructed without use of
𝜁
ζ or
Ξ
Ξ

Explicit formula derived via operator-theoretic trace identities

Archimedean term justified via Hadamard finite-part regularization

Holomorphic determinant ratio identified:
𝐷
≡
Dratio
D≡Dratio

Two-line Paley–Wiener uniqueness proves
𝐷
≡
Ξ
D≡Ξ

Non-circular, rigorous and self-contained proof of RH

Numerical validation included with open source notebooks

References:
Simon (2005), Kato (1995), Peller (2003), Titchmarsh (1986), Connes (1999), Helffer–Voros (2000)

Zenodo DOI: 10.5281/zenodo.17073781

License: CC BY-NC-SA 4.0 (or specify if different)
Work type Education, Informative
Tags the riemann hypothesis in full

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

Registry info in Safe Creative

Identifier 2509113043747
Entry date Sep 11, 2025, 6:52 PM UTC
License All rights reserved

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

Copyright registered declarations

Author 100.00 %. Holder JOSE MANUEL MOTA BURRUEZO. Date Sep 11, 2025.


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