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2025
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| Online Access: | https://doi.org/10.5281/zenodo.16937789 |
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| author | Loiseau, Robert |
| author_facet | Loiseau, Robert |
| contents | <p>This document is <strong>Chapter 1 of the developing book <em>Spectral Number Theory</em></strong>, presenting the <strong>Spectral Basis of the Integers</strong>.</p> <p>Earlier versions of this upload were titled <em>The Spectral Basis Theorem</em> (SBT). Upon further reflection, I recognized that while the underlying construction is rigorous and important, it is not a <em>new theorem</em> in the sense of establishing a result beyond the Fundamental Theorem of Arithmetic (FTA). Rather, it is a <strong>reinterpretation</strong>: integers can be represented as finite-support prime-exponent vectors, with multiplication corresponding to coordinate addition.</p> <p>Instead of minimizing this idea to a short 2-page “curiosity,” this chapter develops it fully as the <strong>foundational framework</strong> of Spectral Number Theory. It introduces:</p> <ul> <li> <p>The spectral basis representation of integers.</p> </li> <li> <p>Extension to $0$ and $\mathbb{Z}$ via the sign operator.</p> </li> <li> <p>Rational and irrational extensions (roots, transcendentals).</p> </li> <li> <p>Spectral approximations and the real line.</p> </li> <li> <p>Interpretive connections to analysis, signal processing, quantum mechanics, and cryptography.</p> </li> <li>Corrected Figure 1.3 as \vec{a}(4)=(2,0,0,..) (not \vec{a}(4)=(0,2,0,..))</li> </ul> <p>This reframing avoids “over-reaching” claims, but shows that what initially seemed a small notational curiosity — treating prime exponents as coordinates — actually opens a constructive and rigorous path toward a new branch of number theory.</p> <p>Chapter 1 thus serves as the pedagogical and structural foundation for Spectral Number Theory, with later chapters (to appear separately) addressing prime-logarithmic vector spaces, operators, invariants, and connections to analytic number theory.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_16937789 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Number Theory Loiseau, Robert <p>This document is <strong>Chapter 1 of the developing book <em>Spectral Number Theory</em></strong>, presenting the <strong>Spectral Basis of the Integers</strong>.</p> <p>Earlier versions of this upload were titled <em>The Spectral Basis Theorem</em> (SBT). Upon further reflection, I recognized that while the underlying construction is rigorous and important, it is not a <em>new theorem</em> in the sense of establishing a result beyond the Fundamental Theorem of Arithmetic (FTA). Rather, it is a <strong>reinterpretation</strong>: integers can be represented as finite-support prime-exponent vectors, with multiplication corresponding to coordinate addition.</p> <p>Instead of minimizing this idea to a short 2-page “curiosity,” this chapter develops it fully as the <strong>foundational framework</strong> of Spectral Number Theory. It introduces:</p> <ul> <li> <p>The spectral basis representation of integers.</p> </li> <li> <p>Extension to $0$ and $\mathbb{Z}$ via the sign operator.</p> </li> <li> <p>Rational and irrational extensions (roots, transcendentals).</p> </li> <li> <p>Spectral approximations and the real line.</p> </li> <li> <p>Interpretive connections to analysis, signal processing, quantum mechanics, and cryptography.</p> </li> <li>Corrected Figure 1.3 as \vec{a}(4)=(2,0,0,..) (not \vec{a}(4)=(0,2,0,..))</li> </ul> <p>This reframing avoids “over-reaching” claims, but shows that what initially seemed a small notational curiosity — treating prime exponents as coordinates — actually opens a constructive and rigorous path toward a new branch of number theory.</p> <p>Chapter 1 thus serves as the pedagogical and structural foundation for Spectral Number Theory, with later chapters (to appear separately) addressing prime-logarithmic vector spaces, operators, invariants, and connections to analytic number theory.</p> |
| title | Spectral Number Theory |
| url | https://doi.org/10.5281/zenodo.16937789 |