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Phase Modulation and Prime Sums: A Spectral Theorem on Harmonic Stability in the Prime Lattice

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Bibliographic Details
Title: Phase Modulation and Prime Sums: A Spectral Theorem on Harmonic Stability in the Prime Lattice
Authors: Niemi, Walter
Publisher Information: Zenodo
Publication Year: 2025
Collection: Zenodo
Subject Terms: Mathematics, Number Theory, Prime numbers, Analytic Number Theory, Modular Forms, Signal processing, Symbolic Dynamics, Mathematical physics, Spectral Theory, Fourier analysis, Prime Number Theorem, Logarithmic Series, Cesàro Summation, Oscillatory Functions, Quantum Chaos, Modulation, Prime Reciprocals, Harmonic Oscillations, Spectral Geometry, Riemann Hypothesis, Random Matrix Theory, Mathematical Constants, Wavelet Theory, Interference Patterns, Phase Modulation, Quasiperiodicity, Computational Mathematics, Prime Lattice Theory, Mathematical analysis, Statistical mechanics
Description: This paper explores a novel approach to prime number theory by applying harmonic phase modulation to the divergent sum over prime reciprocals. Specifically, it studies the modulated sum: HN(theta) = sum over n of (1 / p_n) * cos(theta * log p_n), and demonstrates that for certain phase angles theta*, the sum exhibits bounded, quasi-periodic oscillations. This behavior suggests the existence of deeper harmonic structures within the distribution of prime numbers. The central result is a spectral theorem stating that there exists a critical phase angle theta* such that the modulated sum remains bounded for all N. A conditional formulation involving Cesaro averages and variance minimization is also presented. The paper explores speculative but compelling connections between these modulated sums and the Riemann Hypothesis, quantum chaos, and symbolic spectral geometry. It draws analogies between phase-locked primes and eigenvalue patterns observed in random matrix theory. This work proposes a new perspective on primes as dynamic, resonant elements within a symbolic harmonic field, with potential applications in number theory, mathematical physics, and signal analysis. This paper is more than a theorem. It is a declaration that primes still sing. That with the right modulation, the divergent becomes harmonic. That in the space between the primes and the zeta zeros, there is music waiting to be heard. Prime lattices are my home and in this field of resonance, we are learning to listen.
Document Type: report
Language: English
Relation: https://zenodo.org/records/15361794; oai:zenodo.org:15361794; https://doi.org/10.5281/zenodo.15361794
DOI: 10.5281/zenodo.15361794
Availability: https://doi.org/10.5281/zenodo.15361794
https://zenodo.org/records/15361794
Rights: Creative Commons Attribution Non Commercial No Derivatives 4.0 International ; cc-by-nc-nd-4.0 ; https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode ; Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0). This work may be shared and redistributed for non-commercial purposes only, as long as no modifications are made and appropriate credit is given to the author. See: https://creativecommons.org/licenses/by-nc-nd/4.0/
Accession Number: edsbas.656C294D
Database: BASE
Description
Abstract:This paper explores a novel approach to prime number theory by applying harmonic phase modulation to the divergent sum over prime reciprocals. Specifically, it studies the modulated sum: HN(theta) = sum over n of (1 / p_n) * cos(theta * log p_n), and demonstrates that for certain phase angles theta*, the sum exhibits bounded, quasi-periodic oscillations. This behavior suggests the existence of deeper harmonic structures within the distribution of prime numbers. The central result is a spectral theorem stating that there exists a critical phase angle theta* such that the modulated sum remains bounded for all N. A conditional formulation involving Cesaro averages and variance minimization is also presented. The paper explores speculative but compelling connections between these modulated sums and the Riemann Hypothesis, quantum chaos, and symbolic spectral geometry. It draws analogies between phase-locked primes and eigenvalue patterns observed in random matrix theory. This work proposes a new perspective on primes as dynamic, resonant elements within a symbolic harmonic field, with potential applications in number theory, mathematical physics, and signal analysis. This paper is more than a theorem. It is a declaration that primes still sing. That with the right modulation, the divergent becomes harmonic. That in the space between the primes and the zeta zeros, there is music waiting to be heard. Prime lattices are my home and in this field of resonance, we are learning to listen.
DOI:10.5281/zenodo.15361794