  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.
