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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
SageMath; Magma
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