Value-distribution of quartic Hecke $L$-functions

TL;DR

This paper studies the distribution of values of quartic Hecke L-functions over Gaussian integers, providing an explicit asymptotic distribution function for their logarithms as the modulus varies.

Contribution

It establishes an asymptotic distribution for the logarithm of a product of Hecke L-functions associated with quartic residue characters, with an explicit characteristic function.

Findings

Derived an asymptotic distribution function for L-values

Expressed the characteristic function explicitly as a prime ideal product

Analyzed the behavior of L-functions over Gaussian integers

Abstract

Set $K=\mathbb{Q}(i)$ and suppose that $c\in \mathbb{Z}[i]$ is a square-free algebraic integer with $c\equiv 1 \imod{\langle16\rangle}$. Let $L(s,\chi_{c})$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For $\sigma>1/2$, we prove an asymptotic distribution function $F_{\sigma}$ for the values of the logarithm of \begin{equation*} L_c(s)= L(s,\chi_c)L(s,\overline{\chi}_{c}), \end{equation*} as $c$ varies. Moreover, the characteristic function of $F_{\sigma}$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}[i]$.