Primes in short intervals: Heuristics and calculations

TL;DR

This paper presents heuristic-based conjectures about the distribution of primes in short intervals of length up to rac{(\u221d x)^2}{2}, supported by existing data, suggesting the maximum number of primes grows slowly.

Contribution

It introduces new heuristic conjectures on prime counts in short intervals and analyzes their consistency with available data.

Findings

Conjectures suggest slow growth of maximum primes in short intervals.

Data provides partial support for the conjectures.

Room remains for modifications based on further evidence.

Abstract

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.