Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers

TL;DR

This paper explores the resonance phenomena in quantum cohomology of Grassmannians and reveals a surprising connection to prime number distribution, providing new formulations related to the Riemann Hypothesis.

Contribution

It establishes a link between coalescence in quantum cohomology and prime number distribution, offering two novel formulations of the Riemann Hypothesis.

Findings

Frequency of coalescence influenced by prime numbers

Equivalent formulations of the Riemann Hypothesis involving Grassmannians

Asymptotic estimates for non-coalescing Grassmannians

Abstract

The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function.