Why should one expect to find long runs of (non)-Ramanujan primes ?

TL;DR

This paper explores the unexpected long runs of Ramanujan and non-Ramanujan primes, providing a heuristic explanation based on the Prime Number Theorem, and discusses the discrepancy between observed and predicted run lengths.

Contribution

It offers a heuristic explanation for the occurrence of long prime runs, highlighting limitations of current predictions and suggesting directions for future research.

Findings

Long runs of RPs and non-RPs are statistically significant.

Heuristic based on Prime Number Theorem explains some phenomena.

Observed longer runs of non-RPs than RPs remain unexplained.

Abstract

Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are statistically significantly longer than one would expect if one was tossing an unbiased coin. In this discussion paper we attempt a heuristic explanation of this phenomenon. Our heuristic follows naturally from the Prime Number Theorem, but seems to be only partly satisfactory. It motivates why one should obtain long runs of both RPs and non-RPs, and also longer runs of non-RPs than of RPs. However, it also suggests that one should obtain longer runs of RPs than have so far been observed in the data, and this issue remains puzzling.