| rxiVerse > Number Theory > 2608 |
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| rxiVerse > Number Theory > 2608 |
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[4] rxiVerse:2608.0059 [pdf] submitted on 2026-08-20 08:29:45
Authors: Zhi Cheng
Comments: 17 Pages.
We propose a discrete nonlocal field model on integer lattices in which the coupling strength between sites is determined by the logarithm of their greatest common divisor (GCD). Prime numbers emerge as weakly connected defect states within a strongly coupled composite network. We demonstrate that the low-lying eigenmodes of the graph Laplacian derived from this coupling exhibit localized response peaks at prime positions, and that a combination of gradient-based and amplitude-depression features captures approximately 70% of primes without explicit primality information. A post-hoc comb filter—mathematically equivalent to a restricted sieve operation on the candidate set—elevates precision to 93—100%. Across four intervals spanning three orders of magnitude, the model achieves F1 scores of 0.75—0.89. We provide rigorous matrix definitions, complete parameter specifications, and ablation studies. The framework offers a physically motivated perspective on prime distribution, suggesting that arithmetic structure can be encoded in spectral properties of nonlocal interaction networks. However, we emphasize that this method is not competitive with classical primality testing algorithms; its value lies in revealing structural connections between number theory and spectral physics. The method performs well in intervals where all primes are isolated nodes in the GCD subnetwork (p>b/2); performance degrades substantially when small primes with multiples are included.
Category: Number Theory
[3] rxiVerse:2608.0019 [pdf] submitted on 2026-08-06 19:56:12
Authors: Felix Reichel
Comments: 18 Pages.
This paper presents a novel simple proof of inequalities for the ABC conjecture using Stirling’s series. The approach bridges Stirling’s approximation with the radical function, producing explicit, computationally testable bounds. Specifically, for any ε > 0, there exists C_ε such that for co-prime a + b = c, c ≤ C_ε·rad(abc)^(1+ε).Furthermore this approach turns an abstract conjecture into verifiable numerical inequalities.
Category: Number Theory
[2] rxiVerse:2608.0012 [pdf] submitted on 2026-08-03 19:52:50
Authors: Yu Pan
Comments: 7 Pages.
We define the Uniform Clustering Conjecture (UC Conjecture) for the non-trivial zeros of the Riemann zeta function. Two mutually independent, fully self-contained analytic approaches are constructed to rigorously establish the UC Conjecture. Starting from the UC property, we derive the weak growth condition for the prime remainder term, the convergence of weighted squared integrals, and finally prove that every non-trivial zero satisfies $Ree(ho)=tfrac12$.All arguments have undergone comprehensive strengthening, including uniform bounds for infinite series, positive lower bounds of $L^2$ integrals, asymptotic separation of oscillatory orders, and elimination of infinite subsequence cancellation risks. textbf{Subject to the validity of all analytic estimates and infinite-interval asymptotic arguments within this paper, the Riemann Hypothesis holds true.
Category: Number Theory
[1] rxiVerse:2608.0008 [pdf] submitted on 2026-08-03 00:43:47
Authors: Yu Pan
Comments: 6 Pages.
Based on the explicit formula for the Riemann $zeta$-function, we reconstruct the fluctuation of the prime-counting remainder as a complex oscillatory field induced by non-trivial zeros. By constructing a dyadic energy integral, we decompose the squared modulus of the oscillatory field into self-coupled diagonal terms and interference cross terms between distinct zeros. We introduce the exponential suppression condition $H_2$ for cross terms and rigorously prove the equivalence between $H_2$ and the uniform repulsion conjecture (UC) for zeros with large real parts.Using zero-density bounds, unconditional estimates of the logarithmic derivative of $zeta(s)$, and Weyl equidistribution theory, we classify all resonance mechanisms capable of breaking the exponential suppression. We unconditionally rule out coherent superposition from zero clusters, leaving only the mechanism of phase-locked close zero pairs.To eliminate the boundary case that classical analytic theory cannot exclude — decoupling between real-part perturbations and imaginary-part perturbations — we apply the local singularity balance principle for the $zeta$-function. We prove unconditionally that perturbations of real and imaginary parts of high real-part zeros satisfy intrinsic scaling rigidity $deltaasympDeltagamma$, removing all extra axiomatic assumptions and closing all logical gaps. Finally we establish $sigma^*le tfrac12$, yielding a purely analytic, unconditional proof of the Riemann Hypothesis.
Category: Number Theory