Mock modular Eisenstein series with Nebentypus

Abstract

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.