Davenport and Erdos, and more recently Lamzouri, have investigated the distribution of short character sums as
varies, for a fixed non-principal character
modulo
. In particular, Lamzouri conjectured that these sums should have a Gaussian limit distribution (real or complex according as
is real or complex) provided
satisfies
but
. I will describe some work in progress in connection with this conjecture. In particular, I will try to explain that the conjecture cannot quite be correct (one need not have Gaussian behaviour for
as large as
), but on the other hand one should see Gaussian behaviour for even larger
for most characters