Value-distribution of cubic Hecke $L$-functions

TL;DR

This paper studies the distribution of values of certain Artin L-functions associated with cubic fields over the imaginary quadratic field , providing explicit distribution functions and applications to Brauer-Siegel and Euler-Kronecker constants.

Contribution

It introduces asymptotic distribution functions for the logarithm and derivative of Artin L-functions in cubic fields, with explicit formulas and applications to number field invariants.

Findings

Established asymptotic distribution functions for L-values

Derived explicit characteristic functions as prime ideal products

Proved distribution results for Brauer-Siegel error terms

Abstract

Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let $\zeta_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$ and $\zeta_k(s)$ be the Dedekind zeta function of $k$. For fixed real $\sigma>1/2$, we obtain asymptotic distribution functions $F_{\sigma}$ for the values of the logarithm and the logarithmic derivative of the Artin $L$-functions \begin{equation*} L_c(\sigma)= \frac{\zeta_{k(c^{1/3})}(\sigma)}{\zeta_k(\sigma)}, \end{equation*} as $c$ varies. Moreover, we express the characteristic function of $F_{\sigma}$ explicitly as a product indexed by the prime ideals of $\mathfrak{O}_k$. As a corollary of our results, we establish the existence of an asymptotic distribution function for the error term of the Brauer-Siegel asymptotic formula for the family of number fields $\{k(c^{1/3})\}_{c}$. We also deduce a similar result for the Euler-Kronecker constants of this family.