expansions for the $G$-function are known.

### Motivation in representation theory

In addition to potential applications in number theory, it is remarkable that the quantities investigated here appear in group theory as well. Given an $n$, recall that the theory of representations of $\mathcal{S}_n$ in characteristic 0 can be fully understood via the action of the group its Specht modules, which form a complete set of irreducible representations, are indexed by partitions of $n$, and actually realized over $\mathbb{Z}$.

One can also define them over a field $k$ of characteristic $p>0$, but things get more complicated as the Specht modules will no longer be simple. Instead, $k\mathcal{S}_n$ will decompose into a sum $\oplus_{i} B_i$ of two-sided ideals, called *$p$-blocks*. One can then decompose $1 = e_1+\cdots + e_r$ in this direct sum (each $e_i$ is central and idempotent). Given a Specht module $V^\lambda$ associated to the partition $\lambda$, we see that only one of the $V^\lambda \cdot e_i$ can be nontrivial. We say that $V^\lambda$ (and by extension the partition $\lambda$) *belongs* to the corresponding block.

Nakayama’s conjecture (now a theorem of Brauer and Robinson) says that two partitions belong to the same $p$-block iff they have the same $p$-cores, 