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.

Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions

“…If there exists a constant c F (0) ∈ C for which F (τ ) − c F (0) decays as τ → i∞, and if a similar condition holds as τ → Q, we define (see [19] for the statement written in this generality) the holomorphic projection of F as…”

Formulas for moments of class numbers in arithmetic progressions

“…Suppose furthermore that F (τ )−P i∞ (q −1 ) has moderate growth, where P i∞ ∈ C[x] and that a similar condition holds as τ → Q. Following Sturm [20] and Gross-Zagier [9, Proposition 5.1, p. 288], we define (see [16] for it written in this generality) the holomorphic projection of F…”

Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler-Selberg type relations

“…The integral is an inverse Laplace transform and can be computed using equation (11) on page 215 of [9]. We obtain 2(4πm…”

“…Further examples could be concluded for other objects whose generating functions are (the holomorphic part of) weight 3 2 harmonic Maass forms, e.g. the level N Hurwitz class numbers, the classical spt partition function of Andrews, or the small divisor functions studied in [10,11]. For the sake of succinctness, we omit them here.…”