Integer partitions, probabilities and quantum modular forms

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.