Ranks, Cranks, and Automorphic Forms

The results of this thesis cover the author’s work on partitions, on statistics related to the rank and crank, and on analytic number theory. The connection between t-cores and sums of squares was initiated by Ono and Sze for 4-cores and further studied by Bringmann, Kane, and Males for self-conjugate 7-cores. Males and the author investigate this phenomenon for all t-cores and self-conjugated t-cores, and the details of this paper and of an alternate proof of a q-series identity of Garvan, Kim, and Stanton are provided in Chapter II. Recently, Stanton conjectured that certain polynomials defined in terms of the rank and crank have divisibility properties that may give an insight into how to define a map between the equinumerous classes provided by the rank and crank. Bringmann, Gomez, Rolen, and the author prove part of Stanton’s conjecture and provide cranks for k-colored partitions (similar to those of Rolen, Wagner, and the author) that appear to satisfy a conjecture similar to Stanton’s. Full proofs of these results are given in Chapter III. The log-concavity of p(n) for n > 25 is a well-known result that was independently proven by Nicolas and by DeSalvo and Pak. Many other inequalities combining multiplicative and additive properties of the partition function and related functions have been shown. Chern, Fu, and Tang and Heim and Neuhauser have conjectured such an inequality for k-colored partitions and fractional partitions respectively, and work of Bringmann, Kane, Rolen, and the author has provided partial progress towards the conjecture of Heim–Neuhauser and proved the conjecture of Chern–Fu–Tang. In particular, it follows that the k-colored partition function is log-concave for integral k ≥ 3. Details of the analytic proof of this fact are given in Chapter IV. In the final two chapters, a summary of the author’s work with Griffin, Ono, Rolen, Thorner, and Wagner concerning effective results towards an alternative criterion for the Riemann Hypothesis and of forthcoming work of Alsharif, Gibson, de Laat, Milinovich, Rolen, Wagner, and the author concerning the proportions of distinct zeros of the Dedekind zeta function are given.