2022/03/01 by Kelly Isham, Isham, Kelly
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2203.00646
openalex publication_date 2022/03/01 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
It is well-known that for each fixed n and e, the number of subgroups of index pe in ℤn is a polynomial in p. Is this true for subrings in ℤn of index pe? Let fn(k) denote the number of subrings of index k in ℤn. We can define the subring zeta function over ℤn to be ζℤnR(s) = ∑k ≥ 1 fn(k)k-s. Is this zeta function uniform? These two questions are closely related. In this paper, we describe what is known about these questions, and we make progress toward answering them in a couple ways. First, we describe the connection between counting subrings of index pe in ℤn and counting the solutions to a corresponding set of equations modulo various powers of p. We then show that the number of solutions to certain subsets of these equations is a polynomial in p for any fixed n. On the other hand, we give an example for which the number of solutions to a certain subset of equations is not polynomial. Finally, we give an explicit polynomial formula for the number of `irreducible' subrings of index pn+2 in ℤn.