Dirichlet Series and Asymptotics for Generalized Legendre Factorials

Abstract

Let $K$ be a fixed number field, let $\hat{I}\pounds$ 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 $\hat{I}\pounds$. We study the ideal-valued factorial defined by $v_{\mathfrak{p}}(n{!}_{K,f,\hat{I}\pounds })={\sum }_{k\ge 0}\lfloor \frac{n}{f(\mathfrak{p})(\mathrm{N}\mathfrak{p}{)}^k}\rfloor$. Assume that $f(\mathfrak{p})=c\mathrm{N}\mathfrak{p}+O((\mathrm{N}\mathfrak{p}{)}^{1-\hat{I}´})$ for some $c>0$ and $\hat{I}´>0$. 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 $$\log {\mathrm{N}}_{\hat{I}}\pounds (n{!}_{K,f,\hat{I}\pounds })=c^{-1}n\log n+C_{K,f,\hat{I}\pounds }n+O(n{e}^{-a\sqrt{\log n}})$$ for some $a>0$. The linear coefficient is explicit: $C_{K,f,\hat{I}\pounds }=c^{-1}({\hat{I}}^3+\hat{I}{º}_{K,\hat{I}\pounds }-\log c-1)+J_{K,f,\hat{I}\pounds }(1)$, where $\hat{I}{º}_{K,\hat{I}\pounds }$ is the constant term in the Laurent expansion of $-\hat{I}{\P }_{K,\hat{I}\pounds }^{\prime }(s)/\hat{I}{\P }_{K,\hat{I}\pounds }(s)$ at $s=1$, and $J_{K,f,\hat{I}\pounds }$ 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.