2022/11/29 by Mishra, Hrishabh, Ray, Anwesh
#05A05 #11A25 #11H06 #11M41 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2211.16595
Let n and k be positive integers, and fn(k) (resp. gn(k)) be the number of unital subrings (resp. unital irreducible subrings) of ℤn of index k. The numbers fn(k) are coefficients of certain zeta functions of natural interest. The function k↦ fn(k) is multiplicative, and the study of the numbers fn(k) reduces to computing the values at prime powers k=pe. Given a composition α=(α1, …, αn-1) of e into n-1 positive integers, let gα(p) denote the number of irreducible subrings of ℤn for which the associated upper triangular matrix in Hermite normal form has diagonal (pα1, …, p^αn-1,1). Via combinatorial analysis, the computation of fn(pe) reduces to the computation of gα(p) for all compositions of i into j parts, where i≤ e and j≤ n-1. We extend results of Liu and Atanasov-Kaplan-Krakoff-Menzel, who explicitly compute fn(pe) for e≤ 8. The case e=9 proves to be significantly more involved. We evaluate fn(e9) explicitly in terms of a polynomial in n and p up to a single term which is conjecturally a polynomial. Our results provide further evidence for a conjecture, which states that for any fixed pair (n,e), the function p↦ fn(pe) is a polynomial in p. A conjecture of Bhargava on the asymptotics for fn(k) as a function of k motivates the study of the asymptotics for gα(p) for certain infinite families of compositions α, for which we are able to obtain general estimates using techniques from the geometry of numbers.