What does this research mean for the field?

The twin prime conjecture can be rigorously reformulated as an exact algebraic and structural problem involving quasimodular divisor-sum convolutions, translating the infinitude of twin primes into a continuous analytic domain. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To reformulate the twin prime conjecture from traditional sieve-theoretic methods into an exact algebraic identity governed by quasimodular divisor-sum convolutions and harmonic analysis.

August 23, 2026Open Access

MacMahon Convolutions and a Divisor-Gap Reformulation of the Twin Prime Conjecture

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Authors

JTJorge Daniel Taraborelli

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Implication

Theoretical analysis demonstrates an exact algebraic reformulation of the twin prime conjecture using quasimodular divisor-sum convolutions, suggesting a novel path via spectral analysis.

Key Points

To reformulate the twin prime conjecture from traditional sieve-theoretic methods into an exact algebraic identity governed by quasimodular divisor-sum convolutions and harmonic analysis.
Applied a quadratic restriction (n = m^2) to MacMahon's divisor identity to formulate the Square Coupling Theorem relating square convolutions to boundary divisor sums.
Constructed a Divisor Gap functional and analyzed its separation bounds using the AM-GM inequality, Ramanujan Fourier expansions, and Dirichlet-type series.
Proved a discrete separation boundary ('Forbidden Zone') where the divisor gap functional vanishes strictly for twin primes and is bounded below by m^(3/2) for composite states.
Identified a modular conservation law modulo 12 for twin prime centers m = 6u, demonstrating why elementary congruence techniques are structurally insufficient.
Reformulated the twin prime conjecture as an open spectral threshold problem and a convergence dichotomy in Dirichlet-type indicator series.

Cite This Study

Jorge Daniel Taraborelli (2026) studied this question.

synapsesocial.com/papers/6a8aae207677a34114446e05 https://doi.org/10.5281/zenodo.22045967

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Implication

Theoretical analysis demonstrates an exact algebraic reformulation of the twin prime conjecture using quasimodular divisor-sum convolutions, suggesting a novel path via spectral analysis.

MacMahon Convolutions and a Divisor-Gap Reformulation of the Twin Prime Conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

MacMahon Convolutions and a Divisor-Gap Reformulation of the Twin Prime Conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

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