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Adelic & p-adic Methods

Studying all primes at once through adèles and local fields.

21 related articles

What this theme is

The adèles package the real numbers together with every p-adic field into a single object, letting one treat all primes simultaneously. Tate's thesis derived the functional equation of ζ(s) adelically, and papers here pursue adelic flows, p-adic operators, and idele-class structures as routes to a spectral realization of the zeros.

Why it recurs

The adelic viewpoint is the natural home of the functional equation and of Connes' spectral interpretation of RH. It recurs whenever an author wants the symmetry s ↔ 1−s to be structural rather than accidental.

Relevance to the Riemann Hypothesis

Connes reformulated RH as a trace formula on an adelic space; positivity of a certain distribution there is equivalent to the Hypothesis. Adelic methods aim to build the Hilbert–Pólya operator in a setting where the functional equation is built in.

Adelic Flows and Spectral Rigidity: Bridging Quantum Chaos and the Critical Line

This article investigates the interdisciplinary connection between the Riemann Hypothesis and adelic dynamical systems, proposing a framework where zeta zero...

Adelic Spectral Analysis and the Critical Line Symmetry in the Riemann Hypothesis

This article examines how the construction of adelic L-operators and the study of spectral flow provide new pathways for verifying the vertical alignment of ...

Arithmetic Heights and the Critical Line: A New Pathway via Local Field Valuations

This research article synthesizes local field valuation theory from arXiv:2101.08631 with analytic number theory to propose a novel framework where algebraic...

Bridging Number Theory and Quantum Chaos through Adelic Spectral Operators

This research explores the connection between the Riemann Hypothesis and spectral operator theory, proposing that zeta function zeros correspond to the eigen...

Decoding the Critical Line through Adelic Spectral Dynamics

This article examines the connection between the Riemann Hypothesis and spectral theory, focusing on the construction of adelic operators and transfer operat...

Diophantine Rigidity and the Critical Line: Prime Factor Growth in Recursive Sequences

This article explores how effective p-adic valuation bounds in recursive sequences establish a Diophantine rigidity that parallels the analytic constraints o...

Exponential Prime Detectors and the Analytic Structure of the Critical Strip

This article explores how exponential limit representations of p-adic valuations from arXiv:hal-00810545 can be transformed into analytic tools for investiga...

Hypergeometric Deformations and the Adelic Geometry of Zeta Zeros

This technical analysis investigates the connection between the Riemann Hypothesis and the weak Mellin transforms of second-degree characters, demonstrating ...

Quantum Graphs and the Riemann Hypothesis: A Spectral Approach

A novel research proposal connecting quantum graph spectral zeta functions to the Riemann Hypothesis through bond-scattering matrices and p-adic necklace gra...

Spectral Geometry and the Arithmetic De Rham Flow: A New Path to the Riemann Hypothesis

This article examines how the construction of an Adelic Laplacian and Arithmetic De Rham Flow in arXiv:mathematics_2601_09411v1 provides a spectral-geometric...

Spectral Operator Dynamics and the Critical Line: A New Path to the Riemann Hypothesis

This article explores the connection between the Riemann Hypothesis and a generalized transfer operator acting on adele rings, as proposed in arXiv:2601.1011...

Spectral Operator Theory and High-Precision Computational Frameworks for the Riemann Hypothesis

This article explores the intersection of spectral operator theory on adelic manifolds and high-precision computational algorithms to analyze the distributio...

Spectral Realizations and Adelic Trace Formulas: A New Path to the Riemann Hypothesis

This article investigates a groundbreaking connection between the Keiper-Li criterion and adelic trace formulas, offering a new spectral perspective on the d...

Spectral Resonance and Adelic Operators: A New Frontier for the Riemann Hypothesis

This article explores the connection between spectral theory and the Riemann Hypothesis, analyzing how adelic Hilbert spaces and self-adjoint operators in ar...

Spectral Symmetry and the Adelic Laplacian: A New Frontier for the Riemann Hypothesis

This research analysis explores the connection between the Riemann Hypothesis and the spectral properties of the adelic Laplacian, focusing on the constructi...

Spectral Zeta Functions of Modular Quantum Networks and the Riemann Hypothesis

We investigate a novel class of spectral zeta functions arising from modular quantum networks, establishing a rigorous correspondence between the non-trivial...

The Adelic Laplacian and the Spectral Geometry of Prime Distribution

This article investigates the spectral realization of the Riemann zeta function through non-commutative Adelic Laplacians as proposed in arXiv:2601.09983v1, ...

The Quantum Geometry of Primes: Scattering Theory and the Critical Line

This article examines the operator-theoretic approach to the Riemann Hypothesis, framing the zeta function as a scattering matrix where zeros correspond to r...

The Spectral Geometry of Prime Numbers and Adelic Transfer Operators

This article synthesizes recent advancements in adelic spectral theory to explore the connection between the Riemann Hypothesis and the eigenvalues of Adelic...

The Spectral Music of Primes: Quantum-Adelic Dynamics and the Critical Line

This research article investigates the connection between Quantum-Adelic Spectral Dynamics and the Riemann Hypothesis, proposing that the critical line acts ...

Unlocking the Critical Line Through Adelic Spectral Decompositions

This analysis examines the connection between adelic Hilbert spaces and the Riemann Hypothesis, exploring how the spectral properties of operators in arXiv:2...