  Moments of the partition rank and crank statistics have been studied for
their connections to combinatorial objects such as Durfee symbols, as well as
for their connections to harmonic Maass forms. This paper proves a conjecture
due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's
conjecture states that the moments of the crank function are always larger than
the moments of the rank function, even though the moments have the same main
asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise
asymptotic estimates for rank and crank moments and their differences.
