Ken Ono’s research while affiliated with University of Virginia and other places

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n ≥ 2 is prime if and only if ( 3 n 3 − 13 n 2 + 18 n − 8 ) M 1 ( n ) + ( 12 n 2 − 120 n + 212 ) M 2 ( n ) − 960 M 3 ( n ) = 0 , where the M a ( n ) are MacMahon’s well-studied partition functions. More generally, for MacMahonesque partition functions M a → ( n ) , we prove that there are infinitely many such prime detecting equations with constant coefficients, such as 80 M ( 1 , 1 , 1 ) ( n ) − 12 M ( 2 , 0 , 1 ) ( n ) + 12 M ( 2 , 1 , 0 ) ( n ) + ⋯ − 12 M ( 1 , 3 ) ( n ) − 39 M ( 3 , 1 ) ( n ) = 0 .

We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(\tau),$ defined by $\lambda=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_{\lambda}(\tau):= G_2(\tau)^{m_1} G_4(\tau)^{m_2}\cdots G_{2k}(\tau)^{m_k}.$ For functions $\phi: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight 2k "partition Eisenstein trace" is the quasimodular form ${\mathrm{Tr}}_k(\phi;\tau):=\sum_{\lambda \vdash k} \phi(\lambda)G_{\lambda}(\tau).$ These traces give explicit formulas for some well-known generating functions, such as the kth elementary symmetric functions of the inverse points of 2-dimensional complex lattices $\mathbb{Z}\oplus \mathbb{Z}\tau,$ as well as the 2kth power moments of the Andrews-Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor.

In his ``lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of q-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of ``partition Eisenstein series'', extensions of the classical Eisenstein series $E_{2k}(q)$ defined by $\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \ \ \longmapsto \ \ \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}.$ For functions \phi : \mathcal{P}\mapsto \C on partitions, the {\it weight 2n partition Eisenstein trace} is \Tr_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). For all t, we prove that U_{2t}(q)=\Tr_t(\phi_u;q) and V_{2t}(q)=\Tr_t(\phi_v;q), where $\phi_u$ and $\phi_v$ are natural partition weights, giving the first explicit quasimodular formulas for these series.

In 2010, G.-N. Han obtained the generating function for the number of size t t hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even t t . If n t ( λ ) n_t(\lambda ) is the number of size t t hooks in a partition λ \lambda and S C \mathcal {SC} denotes the set of self-conjugate partitions, then for even t t we have ∑ λ ∈ S C x n t ( λ ) q | λ | = ( − q ; q 2 ) ∞ ⋅ ( ( 1 − x 2 ) q 2 t ; q 2 t ) ∞ t 2 . \begin{equation*} \sum _{\lambda \in \mathcal {SC}} x^{n_t(\lambda )} q^{\vert \lambda \vert } = (-q;q^2)_{\infty } \cdot ((1-x^2)q^{2t};q^{2t})_{\infty }^{\frac {t}{2}}. \end{equation*} As a consequence, if a t ⋆ ( n ) a_t^{\star }(n) is the number of such hooks among the self-conjugate partitions of n n , then for even t t we obtain the simple formula a t ⋆ ( n ) = t ∑ j ≥ 1 q ⋆ ( n − 2 t j ) , \begin{equation*} a_t^{\star }(n)=t\sum _{j\geq 1} q^{\star }(n-2tj), \end{equation*} where q ⋆ ( m ) q^{\star }(m) is the number of partitions of m m into distinct odd parts. As a corollary, we find that t ∣ a t ⋆ ( n ) t\mid a_t^{\star }(n) , which confirms a conjecture of Ballantine, Burson, Craig, Folsom and Wen.

We confirm the speculation that the distribution of t-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length t among the size n self-conjugate partitions is asymptotically normally distributed with mean $\mu_t(n) \sim \frac{\sqrt{6n}}{\pi} + \frac{3}{\pi^2} - \frac{t}{2}$ and variance $\sigma_t^2(n) \sim \frac{(\pi^2 - 6) \sqrt{6n}}{\pi^3}.

Ramanujan’s partition congruences modulo $\ell \in \{5, 7, 11\}$ assert that where $0<\delta _{\ell }<\ell$ satisfies $24\delta _{\ell }\equiv 1\pmod {\ell }.$ By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes $\ell \ge 5,$ we give a new proof of the conclusion that there are infinitely many m for which $p(\ell m+\delta _{\ell })$ is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies $m<(\ell ^2-1)/24,$ representing a significant improvement to the previous bound.

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 natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on ${\mathbb {C}}^2$ . For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty$ . For t -hooks, we prove distributions that are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*}

Recent works at the interface of algebraic geometry, number theory, representation theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of n points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For t-hooks, we prove distributions which are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results which are of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\text {and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right).