Related papers: Mahler Measure, Eisenstein Series and Dimers

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…

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a…

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $\Gamma$ in the Klein bottle $K$. Let $\Gamma_{mn}$ denote the graph obtained by pasting $m$ rows and $n$…

A simple relationship of the form Z_BH = |Z_top|^2 is conjectured, where Z_BH is a supersymmetric partition function for a four-dimensional BPS black hole in a Calabi-Yau compactification of Type II superstring theory and Z_top is a…

We discuss gauge theory with a topological N=2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space $\cal M$ and the partition function equals the Euler number of $\cal M$. We explicitly…

We study a two-dimensional Coulomb gas consisting of a mixture of particles carrying various positive multiple integer charges, confined on a unit circle. We consider the system in the canonical and grand canonical ensembles, and attempt to…

MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with…

We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact…

In this paper we study iterated Eisenstein {\tau}-integrals and multiple Eisenstein L-series, they are functions on the complex upper half plane and form two Q-algebras. They reduce to iterated Eisenstein integrals and multiple Hecke…

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…

This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to…

Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete…

We solve a Lehmer-type question about the Mahler measure of integer-valued polynomials.

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing…

The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as…

We present Euler-type recurrence relations for some partition functions. Some of our results provide new recurrences for the number of unrestricted partitions of $n$, denote by $p(n)$. Others establish recurrences for partition functions…

We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special K\"ahler geometry and…

The purpose of this note is to prove an Euler-type formula for partitions of the M\"obius strip. This formula was introduced in our joint paper with R.~Kiwan, "Courant-sharp property for Dirichlet eigenfunctions on the M\"obius strip"…

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups {\Gamma}_0 (pq) with p and q distinct odd primes, giving an answer to a…

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, $\tau$, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half $\tau$-plane. Two infinite classes…