Paper 125: Prime Corridors and the Arithmetic of Cancellation
Year: 2026
Type: article
Abstract: Can the fluctuations of weighted prime counts be controlled by organizing their interactions across scales? We group primes and prime powers into corridors between consecutive squares and sum squared discrepancies over aligned blocks whose lengths are powers of two. A real sampling argument using the classical explicit formula establishes that bounding this total energy by a constant times the square of the corridor cutoff times the fifth power of the logarithm of that cutoff is equivalent to the Riemann hypothesis. Higher prime powers, centering corrections, and diagonal terms satisfy the required unconditional bounds. An application of Siebert’s sieve controls separations up to the greatest integer not exceeding the fourth power of the logarithm of the corridor cutoff. For longer separations, an established short interval mean square theorem bounds the magnitude of the signed band contribution by a constant times the ambient interval length times the separation scale, divided by any prescribed fixed positive power of the logarithm of the ambient interval length. The ambient interval length is the square of the corridor cutoff minus one. The estimate applies when the separation scale is at least that interval length raised to a fixed exponent strictly between one sixth and one, and at most four less than the interval length. The bounding constant may depend on the chosen exponent and logarithmic power. Exact window energy slopes preserve the signed assembly of these bands. However, a single infinite selection of nonprime integers satisfies the same permitted weights at individual points and the same stated size estimates while defeating every fixed power saving relative to cubic band growth. This construction isolates information about the actual distribution of primes that those estimates do not control. The paper supplies exact identities, specified arithmetic estimates, and a limitation of the overlap method. The global bound at the strength required for the Riemann hypothesis remains unproved. Complete source material and finite checks accompany the derivations.
Author: Sarnowski, Michael
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225649330
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