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The Local-Global Conjecture is False for Generalized Circle Packings
Hanqi Shi, Wenyuan Shi, Ian Whitehead +2
Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the…
Functional equations of axiomatic multiple Dirichlet series, Weyl groupoids, and quantum algebra
Will Sawin, Ian Whitehead
We prove functional equations for multiple Dirichlet series defined by a collection of five geometric axioms. We find functional equations of two types: one modeled on the function…
Domains of Convergence for Polyhedral Packings
Nooria Ahmed, William Ball, Ellis Buckminster +6
Polyhedral circle packings are generalizations of the Apollonian packing. We develop the theory of the Apollonian group, Descartes quadratic form, and related objects for all polyh…
Apollonian Packings and Kac-Moody Root Systems
Ian Whitehead
We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system . We introduce the generating function of a packing, an expo…
On the third moment of II: the number field case
Adrian Diaconu, Ian Whitehead
We establish a smoothed asymptotic formula for the third moment of quadratic {D}irichlet -functions at the central value. In addition to the main term, which is known, we prove…