Integer partitions, probabilities and quantum modular forms

Abstract

What is the probability that the smallest part of a random integer partition of N is odd? What is the expected value of the smallest part of a random integer partition of N? It is straightforward to see that the answers to these questions are both 1, to leading order. This paper shows that the precise asymptotic expansion of each answer is dictated by special values of an arithmetic L-function. Alternatively, the asymptotics are dictated by the asymptotic expansions of quantum modular forms. A quantum modular form is a function on the rational numbers which has pseudo-modular properties and nice asymptotic expansions near each root of unity. This paper contains five examples involving some of the most famous quantum modular forms of Don Zagier. Additionally, this paper contains new generating function identities for the partition questions relevant to this work and three general circle method asymptotics which may be of independent interest. Mathematics Subject Classification: 05A16, 05A17, 11P81, 11P82, 11P84, 60C05 BackgroundLet P(N ) denote the set of all integer partitions of N and write p(N ) = P(N ) for the number of integer partitions of N . The study of the analytic approximation of partition statistics started with Hardy-Ramanujan asymptotic formula [49], further strengthened by Rademacher's exact formula [72]. These formulas give for any positive integer R the asymptoticwhere K =

On the Number of Parts of Integer Partitions Lying in Given Residue Classes

“…The generating function of the individual terms T r,N (n) however has absolutely no modularity properties so that an analogous approach wouldn't work. But using a different version of the Circle Method originally due to E. M. Wright [16] 1 in a convenient formulation given in [15], as well as the famous Euler-Maclaurin summation formula we can prove the following result. Theorem 1.2.…”

“…Wright's Circle Method. In this section, we briefly recall two propositions from [15], based on Wright's version of the Circle Method [4,16], that allow to obtain asymptotic results for products of functions in a fairly general setting.…”

“…For our purposes, we require the following two propositions (see Propositions 1.8 and 1.10 in [15]).…”

The number of parts in certain residue classes of integer partitions

“…We require the following two propositions (Propositions 1.8 and 1.10 in [13]) for our results, the first of which is originally due to Wright [15]. …”

“…A very convenient exposition for our purposes can be found in [4]. We require the following result (see Theorem 1 in [4] and Corollary 4.5 in [13]). …”