Explicit bounds for the number of p-core partitions

• Published in: Transactions of the American Mathematical Society 2014-02, Vol.366 (2), p.875-902
• Main Authors: KIM, BYUNGCHAN, ROUSE, JEREMY
• Format: Article
• Language: English
• Subjects: Adjoints; Coefficients; Discriminants; Fourier coefficients; Generating function; Integers; Interval partitions; Mathematical congruence; Mathematical transformations; Number theory
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-2013-05883-7

In this article, we derive explicit bounds on ct(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 $\mathrm{f}\left(\mathrm{z}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mathrm{e}}_{\mathrm{p}}\mathrm{E}\left(\mathrm{z}\right)\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\underset{\mathrm{i}}{\mathrm{\Sigma }}{\mathrm{r}}_{\mathrm{i}}{\mathrm{g}}_{\mathrm{i}}\left(\mathrm{z}\right)$ 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) := ∑i |ri|, solving a problem raised by Granville and Ono. In the case of general t, we use a combination of techniques to bound ct(n) and as an application prove that for all n ≥ 0, n ≠ t + 1, ct+1(n) ≥ ct(n) provided 4 ≤ t ≤ 198, as conjectured by Stanton.