Summary (taken from Math Reviews):
In his last letter, written in January 1920 to G. H. Hardy, Ramanujan described two families of fifth-order mock theta functions and asserted certain relationships holding in each family. These results were later proved by G. N. Watson [J. London Math. Soc. 11 (1936), 55--80; Zbl 13, 115] and appear also in Ramanujan's ``lost'' notebook together with ten further identities, five for each of the two families. In the paper under review the authors prove that the identities in each family are equivalent, in the sense that if one is true so are all five. Each identity is a complicated relation between $q$-series of a type somewhat similar to those occurring in the Rogers-Ramanujan identities. The proofs use the method of generalized Lambert series developed in an earlier paper of Andrews [Amer. Math. Monthly 86 (1979), no. 2, 89--108; MR 80e:01018].

These results are then related to the two mock theta conjectures, which are expressed in terms of the numbers of partitions of various types. If, as usual, the rank of a partition is defined to be the largest part minus the number of parts, then the first conjecture asserts that $N(1,5,5n)=N(0,5,5n)+\rho\sb 0(n)$, and the second conjecture is of a similar nature. Here $N(b,5,n)$ denotes the number of partitions of $n$ with rank congruent to $b$ modulo $5$, and $\rho\sb 0(n)$ is the number of partitions of $n$ with unique smallest part and all other parts less than or equal to the double of the smallest part. It is shown that the first mock theta conjecture is equivalent to the validity of each of the identities in the first family; a similar result is proved for the second mock theta conjecture and identities in the second family.

A footnote at the end of the paper reports that D. R. Hickerson has now proved the two mock theta conjectures. This establishes the fact, long suspected but hitherto unconfirmed, that the mock theta functions are not merely combinations of theta functions.