Abstract
This thesis concerns the analytic properties of integer weight modular forms and harmonic Maass forms and their interplay with the theory of partitions. In particular, we find new recurrences for the partition counting function p(n), identify subsets of prime numbers using partition-theoretic combinatorial sums, and prove results concerning the zeros of certain harmonic Maass forms for the full modular group.
The analytic theory of partitions dates back to Euler, who obtained the generating function for p(n) as a simple infinite product. In conjunction with his pentagonal number theorem, this gives rise to a straightforward recurrence relation for p(n), which enables calculation of its values for very large n. In thesis, in joint work with Ono, Saad, and Singh, we realize this recurrence as part of a family of recurrences arising from the theory of modular forms. We apply the theory of Rankin–Cohen brackets and the finite dimensionality of modular form spaces to obtain these recurrences, which can be expressed in terms of binomial coefficients, divisor sums, and Fourier coefficients of cusp forms.
In 1920, Percy MacMahon defined an infinite family of q-series which are the generating functions for certain sums involving the part multiplicities of partitions of n. Recent work by Craig, van Ittersum, and Ono showed that infinitely many linear combinations of these series can identify the set of prime numbers; that is, they vanish at n ≥ 2 if and only n is prime. In this thesis, we extend their approach to similarly identify cubes of primes numbers and prime numbers in any fixed arithmetic progression. We prove these results by constructing quasimodular forms of mixed weight which naturally identify these sets; such forms are then written as linear combinations of MacMahon's q-series.
Zeros of modular forms are natural objects of study. Early work by Rankin and Swinnerton-Dyer identified the zeros of Eisenstein series, being located on the upper arc A of the unit circle with real part between 0 and ½ which bounds the bottom of the usual fundamental domain for SL₂(ℤ). In a similar vein, zeros of certain Poincaré series are restricted to this arc, as are modular functions arising from the Hecke action on the j-invariant. In joint work with Ono, this phenomenon was extended to polynomials defined from the Hecke action on a particular weight -10 harmonic Maass form. In this thesis, we realize this fact as a single instance of a broader result for harmonic Maass forms. We make use of the interconnection between Hecke operators and the shadows of harmonic Maass forms and the theory of Maass–Poincaré series to obtain analytic conditions guaranteeing the restriction of zeros to the arc A.
