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[17] ai.viXra.org:2505.0193 [pdf] submitted on 2025-05-29 01:54:22
Authors: Robert Allan
Comments: 8 Pages.
Building upon the geometric Basel construction method established in our previous work[1], we present the discovery of exceptional connections to the Riemann Hypothesisthrough systematic mathematical analysis. The geometric 0.5 offset transformation,previously identified as necessary for Basel convergence, exhibits perfect alignmentwith the Riemann critical line Re(s) = 1/2, representing the first known geometricconstruction to naturally arrive at this fundamental boundary in zeta function theory.Through comprehensive multi-method validation including statistical analysis (R² =0.9854), Fourier spectral correlation (73.9%), uniqueness proofs across 17 alternativeconfigurations, and scale-independent pattern confirmation across grid sizes 36×36through 1152×1152, we demonstrate an overall 81.9% connection score to RiemannHypothesis theoretical framework. The dual mechanism theory—combining 6N boundaryorganization with 0.5 offset transformation—provides geometric intuition for thecritical line's mathematical significance while maintaining systematic convergencetoward π²/6. These findings suggest a profound connection between discrete geometricconstruction and continuous zeta function theory, offering a novel geometricperspective on the most important unsolved problem in mathematics.
Category: Number Theory
[16] ai.viXra.org:2505.0188 [pdf] replaced on 2025-05-31 21:09:20
Authors: Robert Allan
Comments: 5 Pages. (Note by ai.viXra.org Admin: Author's name is required on the article!)
We present a systematic geometric approach to the Basel problem through discrete gridconstruction with constraint-based optimization. Our method generates Basel series terms through sequential N×N grids and demonstrates systematic convergence of geometric ratios toward the Basel constant π²/6. Analysis of six grid scales (36×36 through 1152×1152) reveals consistent exponential improvement in PPL/FPL (potentialprime location/forbidden prime location) ratios, with mathematical projection indicating convergence to π²/6 ≈ 1.6449. This establishes a unified framework where the same mathematical constant governs both infinite series generation and finite geometric optimization.
Category: Number Theory
[15] ai.viXra.org:2505.0134 [pdf] submitted on 2025-05-20 21:51:30
Authors: Hamid Javanbakht
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please cite listed sceintific references)
We propose a reformulation of the Riemann Hypothesis within a higher-categorical, homotopical framework, replacing spectral cohomology with a cohomotopy-theoretic trace formalism. By modeling the spectrum of the Riemann zeta function via flows on an unstable arithmetic space X_Z, we interpret the nontrivial zeros as homotopy classes of fixed points under a Frobenius-type flow. The resulting structure lifts spectral arithmetic topology into a motivic and cyclotomic context, and aligns the Riemann Hypothesis with a confinement theorem in unstable motivic homotopy theory. Connections to Morel—Voevodsky A1-homotopy theory, topological cyclic homology, and arithmetic Tannakian formalism are sketched.
Category: Number Theory
[14] ai.viXra.org:2505.0126 [pdf] submitted on 2025-05-20 21:09:39
Authors: Samuel Bonaya Buya
Comments: 7 Pages.
We present a proof of the Binary Goldbach Conjecture based on the maximal prime gap in an interval and a lower bound on the number of Goldbach partitions. By showing that the maximum prime gap gmax in the interval (0, 2m) is always less than m, we conclude that R(2m) ≥ 1 for all m > 1.
Category: Number Theory
[13] ai.viXra.org:2505.0121 [pdf] submitted on 2025-05-19 03:12:55
Authors: Hamid Javanbakht
Comments: 7 Pages.
We construct a cohomotopical framework for the Birch and Swinnerton-Dyer (BSD) conjecture over number fields, extending spectral methods developed in earlier work. Our approach interprets the order of vanishing and leading coefficient of zeta functions as trace and regulator invariants on an unstable motivic space governed by flow dynamics. By lifting classical cohomological pairings to a homotopical trace formalism, we define a spectral regulator that links the unlinking rank of arithmetic fixed-point flows with the analytic behavior of zeta functions at the central point. This reformulation anticipates a unification of BSD phenomena with flow-stable motivic dualities and sets the stage for a spectral proof in subsequent volumes.
Category: Number Theory
[12] ai.viXra.org:2505.0120 [pdf] submitted on 2025-05-19 03:25:13
Authors: Hamid Javanbakht
Comments: 5 Pages.
This paper develops and partially proves a spectral reformulation of the Birch and Swinnerton-Dyer conjecture for number fields using the motivic cohomotopical frame- work established in previous volumes. We define trace-based regulator spaces arising from unstable fixed-point classes under arithmetic flows and show that, under natural duality and linking constraints, these spaces satisfy an identity matching the order of vanishing of the Dedekind zeta function at its critical point. A determinant pairing constructed over the flow-invariant regulator classes yields a volume form conjecturally equivalent to the leading coefficient. We prove this correspondence in key cases, including totally real fields of rank 1, and for real quadratic fields under the assumption of standard conjectures on motives. The paper also introduces a deformation-theoretic approach to the motivic spectral category, setting the stage for a full proof in the general case.
Category: Number Theory
[11] ai.viXra.org:2505.0119 [pdf] submitted on 2025-05-19 03:56:32
Authors: Hamid Javanbakht
Comments: 7 Pages.
This paper completes the cohomotopical and spectral reformulation of the Birch and Swinnerton-Dyer conjecture over number fields. Building on previous volumes which established the trace-based rank identity and spectral regulator interpretation, we now incorporate torsion subgroups, Tamagawa numbers, and local-global correction factors into the flow-fixed cohomotopy model. We develop a generalized trace pairing formalism, prove a spectral torsion-weighted volume identity, and conjecturally extend the formulation to nonabelian L-functions. The classical components of the BSD formula—rank, regulator, torsion, Tamagawa, and Shafarevich group—are each interpreted geometrically as flow-fixed volumes, trace-null cycles, boundary strata, and duality obstructions in a derived category of flow-equivariant motives. The result is a unified spectral topology of arithmetic that realizes BSD as a motivic trace identity over a regulated flow category, and anticipates a broader categorical Langlands correspondence.
Category: Number Theory
[10] ai.viXra.org:2505.0118 [pdf] submitted on 2025-05-19 04:15:48
Authors: Hamid Javanbakht
Comments: 7 Pages.
This paper extends the spectral arithmetic topology framework into the nonabelian domain by constructing cohomotopical torsors, a spectral étale fundamental group, and a derived category of flow-equivariant motives. We begin by defining torsors as trace-stable homotopy classes in arithmetic flow spaces and develop a homotopical version of the étale fundamental group. Spectral abelianization recovers class field theory from stabilized flow categories. We then introduce a trace-formalism for Hecke stacks and Langlands parameters, and reinterpret automorphic sheaves as objects in a derived flow category. A global trace category is constructed, encompassing torsors, regulators, and flow-invariant sheaves. Within this setting, we prove a nonabelian reciprocity theorem via a categorical pairing between spectral Selmer stacks and trace-stabilized regulators, and establish a Tannakian-style equivalence recovering the spectral fundamental group as a symmetry object. This unifies abelian and nonabelian arithmetic dualities under a spectral and categorical framework, pointing toward a cohomotopical Langlands program grounded in trace geometry.
Category: Number Theory
[9] ai.viXra.org:2505.0106 [pdf] submitted on 2025-05-19 02:43:08
Authors: Marko Vrček
Comments: 2 Pages.
We construct a self-adjoint Dirac operator on a fractal-arithmetic Hilbert space whose spectrum encodes the zeros of the Riemann zeta function. Through explicit decomposition into fractal, arithmetic, and modular components, we achieve spectral correspondence (RMSE 0.11) with the first 100 nontrivial zeros. The construction is formalized as a complete spectral triple, with verified reality structure and summability properties. Numerical diagonalization via restarted Lanczos iteration demonstrates both global distribution matching and local statistics consistent with Montgomery’s pair correlation conjecture.
Category: Number Theory
[8] ai.viXra.org:2505.0105 [pdf] submitted on 2025-05-19 02:39:29
Authors: Marko Vrček
Comments: 4 Pages.
We present a comprehensive theory of structural ratio laws governing prime constellations, with twin primes as the fundamental case. For consecutive twin prime pairs (pn, pn + 2) and (pn+1, pn+1 + 2), the normalized ratio Rn =pn+1+2 2(pn+2) converges exponentially to 0.5 with decay rate b = C2/6, where C2 ≈ 0.66016 isthe twin prime constant. This law generalizes to all admissible prime constellations through Rn(k) = pn+1+k 2(pn+k) → 0.5, exhibiting universal exponential convergence ratestied to respective constellation constants. The deviations Rn−0.5 display f−2 power spectra, revealing deep connections to spectral theory and quantum chaos. These results establish new geometric regularities in prime distributions, transcending classical density and gap approaches.
Category: Number Theory
[7] ai.viXra.org:2505.0103 [pdf] submitted on 2025-05-18 21:17:11
Authors: Hamid Javanbakht
Comments: 7 Pages.
We propose a spectral-geometric framework in which the Riemann Hypothesis is recast as a confinement theorem on the spectrum of a flow operator over an arithmetic cohomology space. This framework, which we call Spectral Arithmetic Topology, constructs a correspondence between prime periodicities and the eigenvalues of a Laplace-type operator acting on the cohomology of a dynamically foliated arithmetic space.
Category: Number Theory
[6] ai.viXra.org:2505.0097 [pdf] replaced on 2025-05-26 23:04:05
Authors: Hans Rieder
Comments: 4 Pages. (Note by ai.viXra.org Admin: For the last time, please cite and list scientific references)
This paper presents a complete and rigorous proof of the Collatz conjecture using a residue class analysis modulo 16. By focusing on the reduced Collatz function, which acts only on odd numbers and skips even intermediate steps, a deterministic structure in the iterative behavior becomes visible. A central contraction lemma identifies conditions under which a true descent occurs.A detailed investigation of all odd residue classes shows that such a descent occurs for every starting number. The proof excludes divergent sequences and non-trivial cycles and shows that every natural number eventually reaches 1. Thus, the conjecture is proven without heuristic arguments, but purelystructurally.
Category: Number Theory
[5] ai.viXra.org:2505.0047 [pdf] replaced on 2025-06-03 20:30:35
Authors: Dobri Bozhilov
Comments: 13 Pages.
The known approximations for the number of prime numbers π(x) include Gauss’s formula x/ln(x) and Riemann’s formula through the logarithmic integral Li(x). The latter is known for its high accuracy but is difficult to compute numerically, as it requires integration.In the present work we propose a new approximation with an elementary structure and exceptionally high precision, which gives much more accurate results compared to Gauss’s formula and almost reaches those of Li.In essence, the new formula represents an improved Gauss formula by turning the logarithm base in the denominator from fixed natural (ln) to one with a floating base.Empirically, very accurate results are established in the range above. For large values, the formula approaches that of Gauss and both become equally accurate, along with Li.
Category: Number Theory
[4] ai.viXra.org:2505.0044 [pdf] submitted on 2025-05-07 20:07:58
Authors: Javier Muñoz de la Cuesta
Comments: 13 Pages.
ABSTRACT. This paper presents a novel quantum vibrational model, rooted in Unified Field Theory (UFT), that proves the Riemann Hypothesis (RH). RH conjectures that all non-trivial zeros of the Riemann zeta function �� (��) have real part �� = 12 . We model these zeros as vibrational modes emerging from a primordial quantum singularity, constructing a Hamiltonian whose eigenvalues correspond to the imaginary parts ���� of the zeros. Through an iterative process involving logical ideation, analytical derivations, numerical simulations, and rigorous refinements, we demonstrate that all non-trivial zeros lie on the critical line Re(��) = 1 2 , resolving RH. The model leverages quantum-inspired concepts such as superposition, entanglement, and iterative state regression,establishes a direct connection with �� (��), and provides a scalable framework that bridgesnumber theory and quantum mechanics.
Category: Number Theory
[3] ai.viXra.org:2505.0028 [pdf] submitted on 2025-05-05 22:05:56
Authors: Lovuyo Melvin Chotelo
Comments: 3 Pages.
We introduce a novel two-index formula for term generation in arithmetic structures. Traditional arithmetic sequences rely on a single index to determine sequence terms. In contrast, our formula leverages the sum of two indices, offering a natural generalization to higher-dimensional structures. This approach maintains linear growth while introducing new symmetry and structural properties, making it potentially valuable for applications in grid-based systems, networks, and combinatorial frameworks.
Category: Number Theory
[2] ai.viXra.org:2505.0013 [pdf] submitted on 2025-05-02 23:59:59
Authors: Viktor Weimer
Comments: 3 Pages. In German (Note by ai.viXra.org Admin: An abstract in the article is required)
This article is based on a systematic consideration of the structure of all natural numbers, with particular attention to powers of 2 andcongruence classes modulo 4, to show that every number enters this structure, either directlyor via transformations, and thus reaches 1 in finite time.
Category: Number Theory
[1] ai.viXra.org:2505.0012 [pdf] submitted on 2025-05-03 00:01:02
Authors: Viktor Weimer
Comments: 3 Pages. In German (Note by ai.viXra.org Admin: An abstract in the article is required)
The Collatz conjecture is algorithmically supported by decomposition into powers of 2. Each component follows a fixed reduction path. The seemingly chaotic process is in fact clearly structured and universally convergent.
Category: Number Theory