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[5] viXra:2606.0031 [pdf] submitted on 2026-06-09 20:59:59
Authors: Jian-Yu Huang
Comments: 27 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
We develop a geometric and spectral framework for studying structural fluctuations in discrete arithmetic sequences, operating entirely within real-variable analysis and without appeal to the Riemann zeta function, Euler products, explicit formulae, or analytic continuation. An arithmetic lattice is mapped onto a transcendental logarithmic manifold via an area-filling projection; the resulting Moiré-type interference field serves as the primary object of study. The principal contributions are as follows. (i) We prove, with explicit constants, that the projection map is injective and that successive images are uniformly separated (Propo-sitions 2.3-2.4). (ii) We establish a uniform L 2 near-orthogonality theorem for finite phase families, yielding a computable operator-norm bound on the associated Gram matrix via Gershgorin's circle theorem [5] (Theorem 3.3). (iii) We introduce an explicit Moiré di-lation operator T σ = −i(x ∂ x + σ) on L 2 ([1, ∞)), show that the geometric phasor family {x −(σ+it) } t∈R constitutes its formal eigenfunction family (Lemma 3.12), and prove by integration by parts that T σ is formally self-adjoint if and only if σ = 1 2 (Theorem 3.14). (iv) We derive quantitative variance bounds for node-count statistics, controlled by the pairwise phase-correlation bounds (Theorem 3.19). Complete exponential-sum estimates with explicit constants (Appendix C) and Abel-smoothing remainder formulae including an incomplete-Gamma bound (Appendix B) are provided. All theoretical claims are either proved using the above tools or explicitly designated as conjectural.
Category: Number Theory
[4] viXra:2606.0030 [pdf] submitted on 2026-06-09 20:46:41
Authors: Payam Danesh
Comments: 12 Pages.
Fibonacci and Lucas sequences are basic examples of second-order recurrences, and their behavior is closely connected to the golden ratio. Bernoulli numbers and special values of the Riemann zeta function also form a classical part of number theory. This paper connects these two areas through exact finite identities. The method starts from exponential generating functions, separates the odd-indexed terms, applies Bernoulli generating functions, and then compares coefficients. This gives a finite formula in which a weighted sum of zeta values at non-positive integers becomes an explicit Fibonacci expression. The same argument also gives a Lucas version, and then extends to every sequence satisfying the Fibonacci recurrence with arbitrary initial values. Exact symbolic checks and residual plot are included to show how the cancellation works. The result is a complete unconditional link between Fibonacci-type recurrences, Bernoulli numbers, and special zeta values.
Category: Number Theory
[3] viXra:2606.0021 [pdf] submitted on 2026-06-06 03:28:18
Authors: Payam Danesh
Comments: 29 Pages.
Polynomial expansions of zeta functions provide a natural way to connect analytic continuation, regularized summation, Mellin analysis, and orthogonal polynomial theory. In this paper we try to develop a shifted Ramanujan—Mellin expansion for the Hurwitz zeta function in the critical strip. The construction combines Abel—Plana regularization over the nonnegative integers, Ramanujan summation for shifted Dirichlet terms, the Cayley transform of the right half-plane, and Mellin transforms of Laguerre functions. The main result proves that the Hurwitz zeta function admits a locally uniformly convergent expansion in a universal polynomial basis that is independent of the shift parameter. The shift appears only through explicit coefficients involving the digamma function and shifted Hurwitz zeta values. The Riemann zeta function is obtained as a special case. On the critical line, the normalized basis forms a complete orthonormal system with respect to a hyperbolic weight, and every zero of each basis polynomial lies on the critical line. The final result gives an exact zero-free compact criterion equivalent to the Riemann Hypothesis.
Category: Number Theory
[2] viXra:2606.0008 [pdf] submitted on 2026-06-03 00:06:14
Authors: Payam Danesh
Comments: 17 Pages.
Ramanujan’s divisor-sum identity gives one of the most analytical positivity arguments in the theory of the Riemann zeta-function: in Ingham’s work it yields the non-vanishing of ζ(s)on the line Rs=1. This paper revisits that mechanism and examines what is required to move it toward the critical strip. We first give a self-contained proof of the Ramanujan—Ingham zero-free line. We then prove that the direct critical-strip analogue fails for a precise Euler-factor reason: the positive Ramanujan square acquires an obstructing pole, while removing that pole destroys positivity already at prime level. This obstruction leads naturally to the Nyman—Beurling Hilbert-space formulation. Using Mellin transforms, we express the relevant closure problem through centered Ramanujan fractional-part functions and derive the exact finite-dimensional Gram system for optimal approximation. We prove fixed-window density of the associated boundary functions and separate the remaining problem into compact approximation and tail control. The main conclusion is a rigorous reduction: within this Ramanujan—Beurling framework, the remaining obstruction to the Riemann Hypothesis is an explicit uniform growing-window approximation estimate with controlled coefficient mass.
Category: Number Theory
[1] viXra:2606.0003 [pdf] submitted on 2026-06-01 20:52:50
Authors: Payam Danesh, Raoul Bianchetti
Comments: 16 Pages.
In this work we offer a careful framework for approaching the critical-line problem associated with the Riemann zeta function. At its heart is a long-standing divide in the subject. On one side are analytic approaches, which study the completed zeta function through its reflection symmetry. On the other side are arithmetic approaches, where related criteria often appear through extreme behavior in divisor functions. The purpose of this paper is not to claim a proof of the Riemann Hypothesis, but to place these two perspectives into a clearer and more usable relationship. The argument begins with reflected analytic data for the completed zeta function. It shows that such data can be described through an odd analytic perturbation, giving a more organized way to understand the analytic side of the problem. This also resolves a common point of confusion: the full complex defect is not required to vanish on the critical line. What matters is more subtle. Under a natural real-symmetry condition, the real part of the defect vanishes on the critical line, and this is the feature that becomes useful for the bridge argument. The arithmetic side is built around Ramanujan’s logarithmic divisor profile. The paper establishes the existence and positivity of the relevant extreme scale in the range needed for the proposed connection. These analytic and arithmetic pieces are then brought together through a real bridge functional, made up of a main sign term and a correction term. The main outcome is a conditional criterion for the critical line. If the bridge functional is zero-adapted at the nontrivial zeros, if the real analytic defect satisfies the required one-sided sign condition, and if the correction term remains strictly smaller than the main term, then every nontrivial zero must lie on the critical line. The contribution of this work is therefore structural rather than conclusive. It does not present the Riemann Hypothesis as solved. Instead, it separates what is already established from what still needs to be proved. The key sign law, the domination estimate, and the zero-adaptation identity remain open requirements for any future application of the framework. Its practical value is that it gives researchers a precise checklist for testing whether a proposed analytic or arithmetic strategy can genuinely support a critical-line argument.
Category: Number Theory
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