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The study of partitions with parts separated by parity was initiated by Andrews in connection with Ramanujan's mock theta functions, and his variations on this theme have produced generating functions with a large variety of different modular properties. In this paper, we use Ingham's Tauberian theorem to compute the asymptotic main term for each of the eight functions studied by Andrews.

In 2015, Lovejoy and Osburn discovered twelve $q$-hypergeometric series and proved that their Fourier coefficients can be understood as counting functions of ideals in certain quadratic fields. In this paper, we study their modular and quantum modular properties and show that they yield three vector-valued quantum modular forms on the group $\Gamma_0 (2)$., Comment: 26 pages; v2: Brief comments added. To appear in the Quarterly Journal of Mathematics. v3: Incomplete funding information corrected

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invariants of $3$-manifolds associated with gauge group $\mathrm{SU}(3)$., Comment: 40 pages; v2: Brief comments and clarifications added. To appear in Communications in Contemporary Mathematics