论文标题
$ e_2 $和$ q $ - $ t $ -hook功能的eichler积分
The Eichler integral of $E_2$ and $q$-brackets of $t$-hook functions
论文作者
论文摘要
对于功能，$ f：\ mathcal {p} \ rightArrow \ mathbb {c} $上的bloch和okounkov定义了一个电源系列$ \ langle f \ rangle_q $，这是$ f $的“加权平均值”。作为$ q = e^{2πiz} $中的傅立叶系列，这样的$ q $ - 支架会生成准模块形式的环，而模块化形式则是Dedekind Eta功能的力量。利用Berndt和Han的工作，我们从$$ f_t（λ）构建模块化对象：= t \ sum_ {h \ in \ Mathcal {h} _t _t（λ）} \ frac {1} {1} {1} {1} {h^2}，$$加权总和比$ t $ t $ t $ t $ t $ t $ t $ t $ t的分区挂接数。我们发现$ \ langle f_t \ rangle_q $是$（1-e_2（tz））/24的eichler积分，我们修改为构建符合符合0个模块化属性的函数$ m_t（z）$。结果，非模块傅立叶系列$$ h_t^*（z）：= \ sum_ {λ\ in \ Mathcal {p}} f_t（λ）q^{|λ| - \ frac {1} {1} {24}} {24}}} $$继承权重$ -1/2 $模块化属性。这些足以暗示Chowla-selberg类型的结果，从而概括了以判别$ d <0 $ d <0 $ topter $ t $ d <0 $τ$评估的重量$ k $代数模块化表单是$ω_d^k，$ $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k $。如果我们让$ψ（τ）：= - πi\ left（\ frac {τ^2-3τ+1} {12τ} \ right） - \ frac {\ log（f log（τ）} {2}，$，则$ t = 1 $我们证明了$ t = 1 $ h_1^*（ - 1/τ） - \ frac {1} {\ sqrt {-iτ}} \ cdot h_1^*（τ）
For functions $f: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, Bloch and Okounkov defined a power series $\langle f\rangle_q$ that is the "weighted average" of $f$. As Fourier series in $q=e^{2πi z}$, such $q$-brackets generate the ring of quasimodular forms, and the modular forms that are powers of Dedekind's eta-function. Using work of Berndt and Han, we build modular objects from $$ f_t(λ):= t\sum_{h\in \mathcal{H}_t(λ)}\frac{1}{h^2}, $$ weighted sums over partition hook numbers that are multiples of $t$. We find that $\langle f_t \rangle_q$ is the Eichler integral of $(1-E_2(tz))/24,$ which we modify to construct a function $M_t(z)$ that enjoys weight 0 modularity properties. As a consequence, the non-modular Fourier series $$H_t^*(z):=\sum_{λ\in \mathcal{P}} f_t(λ)q^{|λ|-\frac{1}{24}} $$ inherits weight $-1/2$ modularity properties. These are sufficient to imply a Chowla-Selberg type result, generalizing the fact that weight $k$ algebraic modular forms evaluated at discriminant $D<0$ points $τ$ are algebraic multiples of $Ω_D^k,$ the $k$th power of the canonical period. If we let $Ψ(τ):=-πi \left(\frac{τ^2-3τ+1}{12τ}\right)-\frac{\log(τ)}{2},$ then for $t=1$ we prove that $$ H_1^*(-1/τ)-\frac{1}{\sqrt{-iτ}}\cdot H_1^*(τ)\in \overline{\mathbb{Q}}\cdot \frac{Ψ(τ)}{\sqrt{Ω_D}}.$$
