Dirichlet Series and Asymptotics for Generalized Legendre Factorials

Abstract

Let $K$ be a fixed number field, let $Σ$ be a finite set of nonzero prime ideals of $\mathcal{O}_K$, and let $f$ be a positive integer-valued function on the prime ideals outside $Σ$. We study the ideal-valued factorial defined by $v_{\mathfrak p}(n!_{K,f,Σ})=\sum_{k\geq 0}\left\lfloor \frac{n}{f(\mathfrak p)(\mathrm{N}\mathfrak p)^k}\right\rfloor$. Assume that $f(\mathfrak p)=c\,\mathrm{N}\mathfrak p+O((\mathrm{N}\mathfrak p)^{1-δ})$ for some $c>0$ and $δ>0$. We derive a Dirichlet series for the logarithmic increments and compare its local prime-power sequence with the ordinary prime-ideal von Mangoldt sequence. A prime-ideal theorem and a Dirichlet hyperbola argument then give $$\log \mathrm{N}_Σ(n!_{K,f,Σ})=c^{-1}n\log n+C_{K,f,Σ}n+O(ne^{-a\sqrt{\log n}})$$ for some $a>0$. The linear coefficient is explicit: $C_{K,f,Σ}=c^{-1}(γ+κ_{K,Σ}-\log c-1)+J_{K,f,Σ}(1)$, where $κ_{K,Σ}$ is the constant term in the Laurent expansion of $-ζ'_{K,Σ}(s)/ζ_{K,Σ}(s)$ at $s=1$, and $J_{K,f,Σ}$ is an absolutely convergent prime-ideal correction near $s=1$. The method also applies to factorial ideals of Legendre subsets whose local class numbers have the corresponding geometric form.