Primes in Short Intervals: Heuristics and Calculations

We formulate, using heuristic reasoning, conjectures for the range of the number of primes in intervals of length y around x, where y << (log x)(2). In particular, we conjecture that the maximum grows surprisingly slowly as y ranges from log x to (log x)(2). Wewill exhibit the available data, showing that it somewhat supports our conjectures, though not so well that there may not be room for some modifications.

Automated summary

This article formulates conjectures for the number of primes in intervals of length y around x using heuristic reasoning. The maximum growth rate of the number of primes is found to be surprisingly slow as y ranges from log x to (log x)(2). The provided data somewhat supports these conjectures, but there may be room for modifications.