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.