Densities of bounded primes for hypergeometric series with rational parameters

TL;DR

This paper derives a formula for the density of primes where hypergeometric series with rational parameters are p-adically bounded, proves the rarity of large densities in certain cases, and discusses open problems for the general case.

Contribution

It establishes a formula for the Dirichlet density of bounded primes for hypergeometric series and proves the rarity of large densities in specific prime denominator cases.

Findings

Derived a formula for the Dirichlet density of bounded primes.

Proved the density is small for parameters with denominators of a special form.

Identified open problems for the general case.

Abstract

The set of primes where a hypergeomeric series with rational parameters is $p$-adically bounded is known by [10] 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 prove this conjecture for hypergeometric series whose parameters have denominators equal to a prime of the form $p = 2q^r + 1$, where $q$ is an odd prime, by establishing an upper bound on the density of bounded primes in this case. The general case remains open. This paper is the output of an undergraduate research course taught by the first listed author in the winter semester of 2018.