Explicit bounds for the number of $p$-core partitions

Abstract:

In this article, we derive explicit bounds on $c_{t} (n)$, the number of $t$-core partitions of $n$. In the case when $t = p$ is prime, we express the generating function $f(z)$ as the sum \[ f(z) = e_{p} E(z) + \sum _{i} r_{i} g_{i}(z) \] of an Eisenstein series and a sum of normalized Hecke eigenforms. We combine the Hardy-Littlewood circle method with properties of the adjoint square lifting from automorphic forms on $\rm {GL}(2)$ to $\rm {GL}(3)$ to bound $R(p) := \sum _{i} |r_{i}|$, solving a problem raised by Granville and Ono.

In the case of general $t$, we use a combination of techniques to bound $c_{t}(n)$ and as an application prove that for all $n \geq 0$, $n \ne t+1$, \[ c_{t+1}(n) \geq c_{t}(n) \] provided $4 \leq t \leq 198$, as conjectured by Stanton.