Number Theory

Spectral Analysis of Prime Distribution via Hankel Operators and the Pentagonal Balance of Logarithmic Sums

Authors: Dhayaa Hussein Razzaq

This research establishes a rigorous algebraic and spectral framework for studying prime distribution. The paper proves that the primality criterion c_n = -mu(text{rad}(n))varphi(text{rad}(n))—which is the additive inverse of OEIS A063659—generates the meromorphic ratio F(s) = -zeta(s)/zeta(s-1) via a Dirichlet series. We construct a self-adjoint Hankel operator M and derive an exact trace identity Tr(M) = L(2i_0 - 1) linking it to the logarithmic derivative of the Riemann zeta function. Furthermore, the "Pentagonal Balance" is presented as a structural equilibrium of five exact algebraic identities for logarithmic sums over primes. The entire mathematical architecture is verified numerically with 50-digit precision and generalized to Dirichlet characters. This work was assisted by Google Gemini (LLM) for LaTeX formatting and structural suggestions

Comments: 9 Pages.

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Submission history

[v1] 2026-04-02 20:51:07

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