关于对数的格点推广和戴德金η函数的变形
On a lattice generalisation of the logarithm and a deformation of the Dedekind eta function
摘要：
我们考虑由Terry Gannon提出的戴德金η函数的变形$E_{L,Λ}^{(m)}(it)$，它依赖于两个$d$维单格$(L,Λ)$和两个参数$(m,t)\in (0,\infty)$.我们证明了在具有固定密度的格空间中，格θ函数的最小值是$E_{L,Λ}^{(m)}(it)$的最大值。证明是基于对对数的格推广的研究，称为格对数，也由特里·甘农定义。我们还证明了自然对数是一类一维格对数上的变分问题的特征。
英文摘要：
We consider a deformation $E_{L,Λ}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,Λ)$ and two parameters $(m,t)\in (0,\infty)$, initially proposed by Terry Gannon. We show that the minimizers of the lattice theta function are the maximizers of $E_{L,Λ}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalization of the logarithm, called lattice-logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterized by a variational problem over a class of one-dimensional lattice-logarithm.
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