arXiv subjects

Pedro Caceres

Publications and source records attributed to Pedro Caceres.

5 recordsLinked to original sources

arXiv2024

If our chaotic operator is derived correctly, then the Riemann hypothesis holds true

This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis. Starting from the explicit zero-counting function $N(T)$, we construct a discrete map on the critical line and analyse its Lyapunov exponents and bifurcation diagrams, showing that the smooth von Mangoldt term generates a strongly unstable flow that captures the global growth of the zero density. Motivated by this dynamics, we define a self-adjoint ``chaotic'' operator $\mathcal{O}_\alpha$ on a weighted Hilbert space with weight $\mathrm{d}N/\mathrm{d}T$, prove its unboundedness and essential self-adjointness, and describe its spectral resolution via the spectral theorem. Finite-dimensional truncations of $\mathcal{O}_\alpha$ yield Hermitian random matrices whose eigenvalue statistics agree numerically with Gaussian unitary ensemble predictions and show qualitative similarities to both Odlyzko's zeta zeros and the hydrogen-atom spectrum, suggesting that $\mathcal{O}_\alpha$ lies in the same universality class as the nontrivial zeros and providing a concrete Hilbert--P\'olya--type framework rather than a proof of the conjecture.

math.GM↗

arXiv2023

Analyzing Dynamical Systems Inspired by Montgomery's Conjecture: Insights into Zeta Function Zeros and Chaos in Number Theory

In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(\pi u)}{\pi u} \right)^2 + \delta(u)$. The recurrence $x_{n+1} = 1 - \left( \frac{\sin(\pi/x_n)}{\pi/x_n} \right)^2 + \frac{1}{x_n}$ emulates eigenvalue repulsion as a quantum operator analogue realizing the P\'olya-Hilbert conjecture. Bifurcation analysis and Lyapunov exponents reveal quantum-like chaos: near $x=0$, linearized dynamics $f(x) = 1 - \pi^2 x^2$ yield Gaussian Lyapunov function $V(x) = C_1 e^{-\pi^2 x^3/3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth $\lambda_n \to \ln(\pi^2/6)$. Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual $\gamma_n$ achieves errors $<10^{-100}$, while spectral density $\rho(E) \sim \frac{\log E}{2\pi}$ matches zeta zero statistics. This bridges Montgomery pair correlation to quantum chaos, providing computational evidence for Riemann zero spacing distributions and supporting the quantum operator hypothesis for $\zeta(1/2+it)$.

math.GM↗

arXiv2022

Chaos Analysis in the Hybrid Quintic Duffing-Riemann Zeta System via Decomposition

This paper presents a comprehensive analysis of the driven cubic-quintic Duffing oscillator \[ \ddot{\phi}+\frac{1}{q}\dot{\phi}+\phi^3+\phi^5=A\cos(\omega t), \] advancing both analytical and numerical chaos theory. Using Melnikov analysis on explicit homoclinic orbits \[ \phi_0(t) = 1-\tanh(t)-\tanh^2(t) \quad \text{and} \quad \phi_0(t) = {\rm sech}_{\rm RZ}(t) -{\rm sech}_{\rm RZ}^2(t),\] we rigorously predict transverse homoclinic intersections and limit cycle bifurcations surrounding the hyperbolic saddle $(0,0)$, establishing chaos onset at $A_\mathrm{chaos}\approx0.34$. A groundbreaking contribution introduces the hybrid quintic Duffing-Riemann zeta system $\ddot{\phi}+\phi^3+\phi^5=A\cos(\omega t)+\Re[\zeta(s)]$, where $\zeta(s)=X(s)-Y(s)$ via C-transformation decomposition. Bifurcation portraits reveal zeta perturbation delays chaos by $24\%$ ($A_\text{chaos}\approx0.42$) while enhancing Lyapunov exponents by $27\%$ ($\lambda_\text{max}=0.14>0.11$). Nontrivial zeros $s_k=1/2+it_k$ emerge as chaos suppressors through entropy-matching $|X(s_k,n)|^2=|Y(s_k,n)|^2$. We prove nontrivial zeros manifest as global Lyapunov minimizers $\lambda(s_k)=\min_{\sigma\in[0,1]}\lambda(\sigma+it_k)$, reformulating the Riemann Hypothesis as a verifiable bifurcation prediction. The unperturbed Hamiltonian $H=\frac{1}{2}\dot{\phi}^2+\frac{1}{4}\phi^4+\frac{1}{6}\phi^6$ and stochastic extensions for biomedical applications are analyzed, positioning number-theoretic chaos control as a novel paradigm bridging nonlinear dynamics and analytic number theory.

math.GM↗

arXiv2022

Iterating sum of power divisor function and New equivalence to the Riemann hypothesis

This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0 \pmod m$. Perfect and multiperfect numbers remain extremely rare; odd perfect numbers are still unknown and must be astronomically large. Here, the emphasis is on the dynamical and statistical structure of the iterates rather than on isolated examples. Three main results are obtained. First, it is proved that no integer $m>1$ can satisfy $\sigma_k(m) \equiv 0 \pmod m$ for all $k \ge 0$, thereby ruling out the existence of "metaperfect" numbers and showing that the iteration of $\sigma$ cannot remain permanently trapped in the residue class $0$ modulo $m$. Second, for certain explicit integers such as $m=6,12,24$, the sequence $\sigma_k(m) \bmod m$ is strictly periodic with small period dividing $L=\mathrm{lcm}(e_i+1)$, where the $e_i$ are the prime exponents of $m$. Bifurcation plots and distributional analysis reveal a transition from rigid two-cycle structure to more complex residue dynamics as $m$ increases. Third, a new equivalence with the Riemann Hypothesis is established: RH holds if and only if, for every even non-squarefree $m \ge 5041$ containing a prime fifth power, \[ \frac{\sigma_k(m)}{\sigma_{k-1}(m)\log\log\sigma_{k-1}(m)} \le e^\gamma, \] and the sequence $\sigma_k(m) \bmod m$ is eventually periodic, uniformly in $k \ge 0$. Extensive computations support these periodicity phenomena, yield non-normal discrete distribution models for the residues, and suggest a connection with a newly proposed Schrodinger-type "Caceres" operator whose spectrum numerically reproduces key statistical features of the nontrivial zeros of the Riemann zeta function.

math.GM↗

arXiv2021

On Congruences for Iterates of the Sum--Power Divisor Function and Conditional Implications for the Riemann Hypothesis

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every $n \leq 400$ there exists $k$ such that $\sigma^k(n) \equiv 0 \pmod{n}$ (where $\sigma^k$ denotes the $k$-fold iteration of the sum-of-divisors function), this paper resolves their reverse question negatively: no integer $n > 1$ satisfies $\sigma^k(n) \equiv 0 \pmod{n}$ for \emph{all} $k \geq 1$. The proof eliminates prior gaps via Lenstra's density-zero bounds $\sigma_k(m) \ll m / \log\log m$ combined with Robin's RH-equivalent criterion $\sigma(n) < e^\gamma n \log\log n + 0.6483 n / \log\log n$ ($n \geq 5041$), showing universal metaperfect divisibility implies RH-violating $\sigma$ growth or low-lying zeta zeros near $s=1$. Among multiperfect $n$ with prime $L = \mathrm{lcm}(1+e_p : p \mid n)$, only $n=6$ satisfies the congruence for all odd $k$, with Shannon entropy $H(\sigma^k(6) \mod 6) \to \log 2$ reflecting periodic order. We analyze bifurcation phenomena in the dynamics $\sigma^k(n) \mod n$, where high-entropy chaotic residues for other $n$ mirror GUE statistics of zeta zeros ($\sim \log T / 2\pi$ near $s=1/2$, $>41\%$ verified on critical line), contrasting the ordered $n=6$ case. Zero rates near $s=1$ (simple pole) and $s=1/2$ bound iterated $\sigma$ distributions, linking to RH via divisor sums and dynamical bifurcations; we conjecture $n=6$ uniquely achieves odd-$k$ divisibility with small period dividing $L$.

math.NT↗