Ramanujan's “Lost” Notebook VI: The mock theta conjectures

Abstract

1. INTRODUCTION In this paper we shall consider only Ramanujan’s two families of fifth- order mock theta functions. These functions were briefly described in Ramanujan’s last letter to G. H. Hardy [ 11, pp. 354-3551, and G. N. Watson subsequently [ 12, 131 proved all the assertions about these functions contained in the letter. Subsequently the identities Watson proved were greatly generalized in [2], Also the fifth-order mock theta functions were shown to have double series expansions involving indefinite quadratic forms [7]. However, there remains a profound mystery about these functions. Namely, no one, including Ramanujan, has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions. This problem was described in detail by Watson [ 12, p. 2741 and redescribed in [7, pp. 113-l 141. In [S, p. 97, Eqs. (3.5) and (3.6)], a formula from the “Lost” Notebook was pointed out that would, if valid, lead to the establishment of at least some of the fifth-order mock theta functions as truly mock theta functions in the sense of Watson [ 12; p. 78, footnote]. There are indeed ten such identities, live for each of the two families. We have two objects in this paper. First we wish to show that the identities in each family are equivalent (i.e., if one is true all five are true, and if one is false all five are false). To introduce our second objective we recall some basic notions from partition theory [4, p. 142; 8; p. 841. The rank of a partition is the largest