Odd density of the partition function

For a natural number $n$, the partition function $p(n)$ is the number of representations of $n$ as nonincreasing sequence of positive integers whose sum is $n$. A folklore conjecture on the partition function asserts that the density of odd values of $p(n)$ is $1/2$. In general, for a positive integer $t$, let $p_t(n)$ be the $t$-multipartition function and $delta_t$ be the density of the odd values of $p_t(n)$. It is widely believed that $delta_t$ exists. Judge and Zanello framed a conjecture which establishes a striking connection between $delta_t$ and $delta_1$. In this talk we shall give recent progress on this conjecture.