无平方因子层数的布尔沃尔什埃塔单位与艾森斯坦基
Boolean Walsh Eta Units and Eisenstein Bases For Squarefree Levels
浏览论文内容
中文总结 AI 辅助
研究无平方因子层数下，通过布尔沃尔什埃塔单位与艾森斯坦基，利用有限傅里叶变换对角化相关矩阵及映射，得到结构定理并确定诸多常数等，应用于希格纳素数乘积得出相关恒等式和关系。
中文摘要 AI 辅助
设\(M>1\)为无平方因子数且\(D(M)\)为其布尔除数立方体。对于每个布尔特征\(\chi_T\)，我们附上艾塔商\[R_T^{(M)}(\tau)=\prod_{d\mid M}\eta(d\tau)^{\chi_T(d)},\qquad\chi_T(d)=(-1)^{|T\cap\supp(d)|}\]。在无平方因子层数时，除数立方体上的有限傅里叶变换能同时对角化无平方因子利戈扎特尖点阶矩阵、弗里克补、阿特金 - 勒纳对尖点标签的作用以及对数艾森斯坦级数的常数项映射。特别地，对于\(T\neq\varnothing\)，\[\ord_{1/c}R_T^{(M)}=\frac{\Lambda_T^{(M)}}{24}\chi_T(c),\qquad\Lambda_T^{(M)}=\prod_{p\in T}(p - 1)\prod_{\substack{p\mid M\p\notin T}}(p + 1)\]，且形式\(D\log R_T^{(M)}\)构成\(M_2(\Gamma_0(M))\)的艾森斯坦子空间的沃尔什基。结构定理还确定了明确的弗里克常数……作为应用，我们专门研究希格纳素数乘积\[N = 2\cdot3\cdot7\cdot11\cdot19\cdot43\cdot67\cdot163\]。第一个布尔边界给出艾塔归一化的希格纳着色划分乘积，而顶部沃尔什特征给出莫比乌斯艾塔单位恒等式\[D\log R_{\mathcal P}^{(N)}(\tau)=40415760-\sum_{n\geq1}\sigma_1(n^\perp)q^n\]。相同应用给出了倒数划分乘积的代数模单位关系以及\(1/\pi\)的精确弗里克固定对数导数恒等式。
英文摘要
Let $M>1$ be squarefree and let $D(M)$ be its Boolean divisor cube. To each Boolean character $χ_T$ we attach the eta-quotient \[ R_T^{(M)}(τ)=\prod_{d\mid M}η(dτ)^{χ_T(d)}, \qquad χ_T(d)=(-1)^{|T\cap\supp(d)|}. \] At squarefree level, the finite Fourier transform on the divisor cube simultaneously diagonalizes the squarefree Ligozat cusp-order matrix, Fricke complementation, Atkin--Lehner action on cusp labels, and the constant-term map for logarithmic Eisenstein series. In particular, for $T\ne\varnothing$, \[ \ord_{1/c}R_T^{(M)} =\frac{Λ_T^{(M)}}{24}χ_T(c), \qquad Λ_T^{(M)}= \prod_{p\in T}(p-1) \prod_{\substack{p\mid M\\ p\notin T}}(p+1), \] and the forms $D\log R_T^{(M)}$ form a Walsh basis of the Eisenstein subspace of $M_2(Γ_0(M))$. The structural theorem also determines explicit Fricke constants, Atkin--Lehner eigenvalues, good-prime Hecke eigenvalues, local $U_p$ triangular blocks, a simultaneous bad-prime eigenbasis, and the indices of two explicit principal cuspidal divisor sublattices inside the formal degree-zero cusp-divisor lattice. As an application we specialize to the Heegner prime product \[ N=2\cdot3\cdot7\cdot11\cdot19\cdot43\cdot67\cdot163. \] The first Boolean boundary gives the eta-normalized Heegner-coloured partition product, while the top Walsh character gives the Möbius eta-unit identity \[ D\log R_{\mathcal P}^{(N)}(τ)=40415760- \sum_{n\ge1}σ_1(n^\perp)q^n. \] The same application gives algebraic modular-unit relations for the reciprocal partition product and exact Fricke-fixed logarithmic derivative identities for $1/π$, interpreted through the modular completion of $E_2$ and through accelerated paired products.
