Primes in short intervals: Heuristics and calculations

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.

arXiv comment37 pages, 12 figures, added a dedication, changes to appendix C. The results are unchanged