One of the most romantic stories in the history of mathematics is that of the friendship between Hardy and Ramanujan. It began and ended with two famous letters. The first, sent by Ramanujan to Hardy in 1913, presents its author as a penniless clerk in a Madras shipping office who has made some discoveries that “are termed by the

Abstract. We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews’ smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions. The series N2v(0, 0; q), defined for v ≥ 1 by

We present fermionic quasi-particle sum representations consisting of a single fundamental fermionic form for all characters of the logarithmic conformal field theory models with central charge cp,1, p ≥ 2, and suggest a physical interpretation. We also show that it is possible to correctly extract dilogarithm identities.

always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The

One of the current major targets in physics is the understanding of integrable quantum field theories in two dimensions. In theories with diagonal scattering matrix there is a convincing conjecture for the spectrum of the Hamiltonian. Suppose that there are K species of particles with masses ma, a = 1,...,K, and a

Characters of rational vertex operator algebras (RVOAs) arising, e.g., in 2- dimensional conformal field theories often belong (after suitable normalization) to the (multiplicative) semi-group E + of modular units whose Fourier expansions are in 1 + q Z≥0[q], up to a fractional power of q. If, furthermore, all characters of a RVOA share this property then we have an example of what we call modular sets, i.e. finite subsets of E + whose elements (additively) span a vector space which is invariant under the usual action of SL(2, Z). The appearance of modular sets is always linked to the appearance of other interesting phenomena. The first nontrivial example is provided by the functions appearing in the two classical Rogers-Ramanujan identities, and generalizations of these identities known from combinatorial theory yield further examples. The classification of modular sets and RVOAs seems to be related. This article is a first step towards the understanding of modular sets. We give an explicit description of the group of modular units generated by E +, we prove a certain finiteness result for modular sets contained in a natural semi-subgroup E ∗ of E +, and we discuss consequences, in particular a method for effectively enumerating all modular sets in E∗.