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Counting finite index subrings of \(\mathbb{Z}^n\). (English) Zbl 07324315
Summary: We count subrings of small index of \(\mathbb{Z}^n\), where the addition and multiplication are defined componentwise. Let \(f_n(k)\) denote the number of subrings of index \(k\). For any \(n\), we give a formula for this quantity for all integers \(k\) that are not divisible by a 9th power of a prime, extending a result of Liu.
MSC:
20E07 Subgroup theorems; subgroup growth
11H06 Lattices and convex bodies (number-theoretic aspects)
11M41 Other Dirichlet series and zeta functions
Software:
SageMath; Magma
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