Densities of bounded primes for hypergeometric series with rational parameters

The set of primes where a hypergeometric series with rational parameters is p -adically bounded is known by Franc et al. (J Number Theory 192:197–220, 2018) to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density to be large. We p...

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Bibliographic Details
Published in:Research in number theory 2020-06, Vol.6 (2), Article 19
Main Authors: Franc, Cameron, Gill, Brandon, Goertzen, Jason, Pas, Jarrod, Tu, Frankie
Format: Article
Language:English
Subjects:
Density
Dirichlet problem
Mathematics
Mathematics and Statistics
Number Theory
Parameters
Upper bounds
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Summary:The set of primes where a hypergeometric series with rational parameters is p -adically bounded is known by Franc et al. (J Number Theory 192:197–220, 2018) to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density to be large. We provide evidence for this conjecture for hypergeometric series 2 F 1 ( x / p , y / p ; z / p ) , with p a prime of the form p ≡ 3 ( mod 4 ) , by establishing an upper bound on the density of bounded primes in this case.
ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-020-00194-1