Abstract
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 \(_2F_1(x/p,y/p;z/p)\), with p a prime of the form \(p\equiv 3\pmod {4}\), by establishing an upper bound on the density of bounded primes in this case.
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