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 provide evidence for this conjecture for hypergeometric series 2F1(x/p,y/p;z/p)$$_2F_1(x/p,y/p;z/p)$$, with p a prime of the form p≡3(mod4)$$p\equiv 3\pmod {4}$$, by establishing an upper bound on the density of bounded primes in this case.