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The aim is to establish a spectral realization of the Riemann-von Mangoldt law through a bounded divisor operator.

August 6, 2026Open Access

Spectral Realization of the Riemann-von Mangoldt Law via a Bounded Divisor Operator

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The aim is to establish a spectral realization of the Riemann-von Mangoldt law through a bounded divisor operator.
Defined the bounded Toeplitz-Catalan operator T on the prime-exponent space.
Mapped trace coefficients to the unbounded inverse operator T−1 to derive spectral density.
Utilized a localized Fourier transform dualization to derive a closed-form formula for mean prime gaps.
Successfully reproduced the Riemann-von Mangoldt law for zeros of the Zeta function via spectral parameter tk.
Demonstrated that the trace of T−1 contains the exact oscillatory contribution of the primes.
Established a deterministic relationship between the distribution of primes and the algebraic structure of the divisor network.

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Demonstrates a spectral realization of the Riemann-von Mangoldt law using a bounded divisor operator, highlighting prime distribution implications.

YOUSEF MUHAMMAD ALSAGHIR Al YOUSEF (2026) studied this question.

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Demonstrates a spectral realization of the Riemann-von Mangoldt law using a bounded divisor operator, highlighting prime distribution implications.