Moments of non-normal number fields -- II

TL;DR

This work analyzes discrete and continuous moments of the Dedekind zeta function for arbitrary number fields by expressing $M_K^{(l)}(T)=\sum_{m\le T} a_K^l(m)$ and $I_K^{(l)}(T)=\int_{1}^{T}|\zeta_K(\tfrac12+it)|^l dt$. The discrete moments have the asymptotic $M_K^{(l)}(T)\sim c(l,K)\,T\log^{m_l}(T)$ with $m_l=\frac{1}{|G|}\sum_{g\in G}\chi_{\rho_H}^l(g)-1$, derived via a Dirichlet-series $D_l(s)$ and Delange–Ikehara. For continuous moments, lower bounds of the expected order are established unconditionally in the non-Galois case using Akbary–Fodden and Chebotarev, namely $I_K^{(l)}(T)\gg T\log^{\beta_K l^2}(T)$ with $\beta_K=|H\setminus G/H|$, and the second-moment bound is slightly sharpened in the non-Galois setting by improved off-diagonal control. The results unify and extend prior work on moments of $\zeta_K$ and connect to Selberg-type conjectures and Langlands reciprocity, offering unconditional growth rates and conditional refinements in the non-normal case.

Abstract

Suppose $K$ is a number field and $a_K(m)$ is the number of integral ideals of norm equal to $m$ in $K$, then for any integer $l$, we asymptotically evaluate the sum \[ \sum_{m\leqslant T} a_K^l(m) \] as $T\to\infty$. We also consider the moments of the corresponding Dedekind zeta function. We prove lower bounds of expected order of magnitude and slightly improve the known upper bound for the second moment in the non-Galois case.

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Conjecture : Conjecture A
• Conjecture : Conjecture B
• Theorem 3
• Proposition 1
• Remark
• Lemma 1
• Lemma 2
• proof