TITLE: Equidistribution of the partition function
QUESTION [13 upvotes]: Experiments with the partition function $p(n)$ (for $n = 1, \dotsc, 100000$) seem to indicate that $p(n)$ is uniformly distributed modulo $q$ for every small prime $q.$ Is this known? Is it a special case of some general conjecture?
EDIT
As pointed out by him whose name cannot be pronounced in the comments, the above is not quite true:
For $q=5,$ the density of $0$ is twice as high as that of other residues (which are equally likely).
For $q=7,$ all nonzero residues are equally likely, but $0$ is more than twice as likely as the others.
For $q=11,$ the residue $0$ is exactly twice as likely as the others (all of which are equally likely).

REPLY [15 votes]: I believe this is not known. Ahlgren and Boylan proved in 2003 that $p(n)$ hits every mod $q$ residue class infinitely often, assuming $q\neq 3$ is a prime. They also gave a lower density estimate for $q\geq 5$, but this is too weak to imply actual positive lower density (except for the zero residue class).