Purpose: Almost 40 years ago, H. Cohen formulated a conjecture about the modularity of a certain infinite family of functions involving the generating function of the Hurwitz class numbers of binary quadratic forms. Methods: We use techniques from the theory of modular, mock modular, and Jacobi forms. Result: In this paper, we prove a slight improvement of Cohen's original conjecture. Conclusions: From our main result, we derive so far unknown recurrence relations for Hurwitz class numbers.

Using holomorphic projection, we work out a parametrization for all relations of products (resp. Rankin-Cohen brackets) of weight 3 2 mock modular forms with holomorphic shadow and weight 1 2 modular forms in the spirit of the Kronecker-Hurwitz class number relations. In particular we obtain new proofs for several class number relations among which some are classical, others are relatively new. We also obtain similar results for the mock theta functions.

In a recent important paper, Hoffstein and Hulse [14] generalized the notion of Rankin-Selberg convolution L-functions by defining shifted convolution L-functions. We investigate symmetrized versions of their functions, and we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and "mixed mock modular" forms.

In this paper, we prove the existence of an infinite-dimensional graded supermodule for the finite sporadic Thompson group Th whose McKay-Thompson series are weakly holomorphic modular forms of weight 1 2 satisfying properties conjectured by Harvey and Rayhaun.

Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.

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By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) “small divisors” summatory functions [Formula: see text]. More precisely, in terms of the weight 2 quasimodular Eisenstein series [Formula: see text] and a generic Shimura theta function [Formula: see text], we show that there is a constant [Formula: see text] for which [Formula: see text] is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews [Formula: see text]-function and the “consecutive parts” partition function. Finally, in analogy with Serre’s result that the weight [Formula: see text] Eisenstein series is a [Formula: see text]-adic modular form, we show that these forms possess canonical congruences with modular forms.

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