The number of ideals of $\mathbb{Z}[x]$ containing $x(x-\alpha)(x-\beta)$ with given index

Mitsugu Hirasaka
Pusan National University

Semin Oh
Pusan National University



Content: Let $A$ denote an integral square matrix and $R$ the subring of the full matrix ring generated by $A$. Then $R$ is a free $\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $R$ with given finite index. Thus, the formal Dirichlet series $\zeta_A(s)=\sum_{n\geq 1}a_nn^{-s}$ is well-defined where $a_n$ is the number of ideals of $R$ with index $n$. In this talk we aim to find an explicit form of $\zeta_A(s)$ when $A$ is the adjacency matrix of a graph whose eigenvalues consists of exactly three integers. which are integral.

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