A new look at Pólya's prime gap heuristics

In a very nice paper from 1959, Pólya explored heuristic methods related to prime gaps originally conjectured by Hardy and Littlewood in 1923. Pólya examined heuristics to justify the Hardy-Littlewood conjecture before testing their conjecture by computer program for gaps of size up to 70 and primes p up to 30 million. Following the result of Zhang (2014) we now know that there are an infinite number of primes for some gap with a maximum size of 246. The Hardy-Littlewood and Pólya results suggest a direct relationship between all even gaps, so that confirming the conjectured relationship would prove the twin prime conjecture as well as the broader Polignac conjecture. In this paper we extend Pólya's results to cover gaps up to 256 and extend the computer confirmation for primes up to one trillion. Along the way we revisit the related older results of Polignac, Hardy, and Littlewood, and the newer results of Zhang, Maynard, and others.