On the Coefficients of q-series and Modular Forms

This thesis is on partitions and analytic number theory. In particular, I prove results about statistical properties of partitions, partition inequalities, and facts about special values of coefficients of modular forms. The central methods of this paper are the theory of integer weight modular forms and the circle method.

It is natural to study statistical questions about the parts of partitions. Recently, Beckwith and Mertens proved that the parts of partitions are asymptotically equidistributed among residue classes modulo $t$, but that there is a bias towards the residue classes inhabited by lower positive integers. In this thesis, I prove that the same phenomenon holds for partitions into distinct parts, and I prove that the biases between residue classes holds for $n > 8$. In order to prove these results, I derive explicit error terms for asymptotic estimates involving Euler--Maclaurin summation and utilize Wright's circle method to prove asymptotic formulas approximating the relevant counting functions.

Motivated by work of Dergachev and Kirillov, new work by Coll, Mayers and Mayers explores new connections between partitions and Lie theory via the index of seaweed algebras. This index may be viewed as a statistic on pairs of partitions, and in this light Coll, Mayers, and Mayers conjectured that a peculiar kind of generating function identity related to this new index statistic. Seo and Yee made a significant step towards proving this conjecture by reducing the problem to demonstrating the non-negativity of the coefficients of a certain $q$-series. In this thesis, I complete the proof of this conjecture using Wright's circle method and effective Euler--Maclaurin summation.

Hook numbers of partitions arise naturally from the connection between partitions and the irreducible representations of the symmetric group. I prove results concerning the number of $t$-hooks that appear within partitions. In joint work with Pun, I prove formulas that give the number of partitions of $n$ which have an even or odd number of $t$-hooks, and as a consequence we prove that these counting functions obey a strange distributions law. We prove these results using the Rademacher circle method.

In joint work with Bringmann, Males and Ono, I prove further asymptotic formulas about the distributions of $t$-hooks and Betti numbers in residue classes. We prove that the Betti numbers associated to Hilbert schemes on $n$ points, which naturally add up to the number of partitions of $n$, are equidistributed among residue classes modulo $b$, while equidistribution fails when partitions are divided up based on the residue class of the number of $t$-hooks. These results are proved using both Rademacher-style and Wright-style circle methods. We also use facts about 2-core and 3-core generating functions to prove that certain coefficients vanish in the cases of 2-hooks and 3-hooks.

Since DeSalvo and Pak proved that the partition function is log-concave, the Tur\'an inequalities have been a popular topic within partition theory. These inequalities govern whether certain polynomials constructed from a given sequence of numbers are hyperbolic. In joint work with Pun, I prove that the $k$-regular partition functions satisfy all the Tur\'an inequalities. We prove this using Hagis' formula for the $k$-regular partition functions and a very general criterion for proving Tur\'an inequalities proven by Griffin, Ono, Rolen, and Zagier.

The Atkin-Lehner newforms are extremely important examples of modular forms. Their coefficients are multiplicative, and the values at prime powers are dictated by two-term linear recurrence relations coming from Hecke operators. In joint work with Balakrishnan, Ono, and Tsai, I prove a methodology for identifying which coefficients of certain integer weight newform $f(z)$ are allowed to take on a given odd value. In particular, our method proves that under suitable assumptions, $f(z)$ has only finitely many Fourier coefficients equal to a given odd prime, and we give an algorithm which determines the possible locations of these prime values by computing integer points on algebraic curves with large genus.