Explicit bounds for the number of p-core partitions

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_pE(z) +Σ_ir_ig_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 GL(2) to GL(3) to bound R(p):= Σ_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 ≥ 0, n ≠t + 1, c_t+1(n) ≥ c_t(n) provided 4 ≤ t ≤ 198, as conjectured by Stanton.