Are there any formulas due to Ramanujan that have still not been proved—or disproved?
If so, what are they?
I believe this conjecture is due to Ramanujan and still open: if $x$ is a real number and $2^x$ and $3^x$ are both integers then $x$ is an integer. There may be other open conjectures due Ramanujan. However, right now I'm mainly interested in formulas, i.e. identities, that he wrote down.