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 σψsm(n). More precisely, in terms of the weight 2 quasimodular Eisenstein series E2(τ) and a generic Shimura theta function 𝜃ψ(τ), we show that there is a constant αψ for which εψ+(τ) := α ψ ⋅E2(τ) 𝜃ψ(τ) + 1 𝜃ψ(τ)∑n=1∞σ ψsm(n)qn is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews spt-function and the “consecutive parts” partition function. Finally, in analogy with Serre’s result that the weight 2 Eisenstein series is a p-adic modular form, we show that these forms possess canonical congruences with modular forms.

中文翻译：

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模拟模块化 Eisenstein 系列与 Nebentypus

根据爱森斯坦级数理论，各种除数函数的生成函数以模形式出现。很自然地要问，在模拟模形式的理论中是否系统地出现了进一步的除数函数。我们使用 Zagier 和 Zwegers 的全纯投影方法确定，对于某些（扭曲的）“小除数”求和函数确实是这种情况σψsm(n). 更准确地说，就权重 2 拟模 Eisenstein 级数而言乙2(τ)和一个通用的 Shimura theta 函数𝜃ψ(τ), 我们证明有一个常数αψ为此 εψ+(τ) ：= α ψ ⋅乙2(τ) 𝜃ψ(τ) + 1 𝜃ψ(τ)∑n=1∞σ ψsm(n)qn 是一个半整体重量（极地）模拟模块形式。这些包括组合对象的生成函数，例如 Andrewssp吨-函数和“连续部分”分区函数。最后，与 Serre 的结果类似，权重2爱森斯坦级数是p-adic 模形式，我们证明这些形式与模形式具有规范同余。