2 citations · 3 across the 14 of their papers we have counts for
17 papers
Quasimodular Forms, Partitions, and Primes in Intervals
Kevin Allen, William Craig
Recent developments in theories of partitions, quasimodular forms, multiple zeta values and -multiple zeta values have led to connections between partitions and primes. Central…
Universal Ramanujan-type congruences for prime-detecting quasimodular forms
Bailey Chrobak, William Craig
We establish the existence of infinitely many Ramanujan-type congruences which hold uniformly for the full -module of prime detecting quasimodular forms. In particular, we prov…
On the distribution of the norm of partitions
Walter Bridges, William Craig
The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note…
The Riemann Hypothesis for period polynomials of cusp forms
William Craig, Wissam Raji
We consider the period polynomials associated with cusp forms of weight on all of , which are generating functions for the…
Hook length biases and general linear partition inequalities
Cristina Ballantine, Hannah Burson, William Craig +2
Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct…
On the coefficients of -series and modular forms
William Craig
In this Ph.D dissertation (University of Virginia, 2022), we prove results about the coefficients of partition-theoretic generating functions and of coefficients of integer weight…
Asymptotic equidistribution for partition statistics and topological invariants
Giulia Cesana, William Craig, Joshua Males
We provide a general framework for proving asymptotic equidistribution, convexity, and log concavity of coefficients of generating functions on arithmetic progressions. Our central…
Seaweed Algebras and the Index Statistic for Partitions
William Craig
In 2018 Coll, Mayers, and Mayers conjectured that the -series is the generating function for a certain parity statistic related to the index of se…
On Ramanujan-type Congruences for Multiplicative Functions
William Craig, Mircea Merca
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overp…
On the Number of Parts in Congruence Classes for Partitions into Distinct Parts
William Craig
For integers , let the function denote the number of parts among all partitions of into distinct parts that are congruent to modulo . We prove…
Distributions on partitions arising from Hilbert schemes and hook lengths
Kathrin Bringmann, William Craig, Joshua Males +1
Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is n…
Distribution properties for t-hooks in partitions
William Craig, Anna Pun
Partitions, the partition function , and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory and representation theory. For…
Variations of Lehmer's Conjecture for Ramanujan's tau-function
Jennifer S. Balakrishnan, William Craig, Ken Ono
We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \…
Variants of Lehmer's speculation for newforms
Jennifer S. Balakrishnan, William Craig, Ken Ono +1
In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of or is a Fourier…
Unique product groups and congruence subgroups
William Craig, Peter A. Linnell
We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. W…
A note on the Higher order Turán inequalities for -regular partitions
William Craig, Anna Pun
Nicolas and DeSalvo and Pak proved that the partition function is log concave for . Chen, Jia and Wang proved that satisfies the third order Turán inequali…
Cubefree Trinomial Discriminants
William Craig
The discriminant of a polynomial of the form has the form when are relatively prime. We investigate when these discriminants…