Polynomial behavior of special values of partial zeta function of real quadratic fields at s=0

TL;DR

This paper investigates the polynomial and quasi-polynomial behavior of special values of partial zeta functions at s=0 for families of real quadratic fields, providing explicit formulas and examples.

Contribution

It demonstrates that the special values of partial zeta functions and Hecke's L-functions at s=0 exhibit quasi-polynomial behavior in certain families of real quadratic fields, with explicit coefficient computations.

Findings

Special values behave as quasi-polynomials under certain conditions

Explicit coefficients of the quasi-polynomials are computed

Examples illustrating the theoretical results are provided

Abstract

We compute the special values of partial zeta function at $s=0$ for family of real quadratic fields $K_n$ and ray class ideals $\fb_n$ such that $\fb_n^{-1} = [1,\delta(n)]$ where the continued fraction expansion of $\delta(n)$ is purely periodic and each terms are polynomial in $n$ of bounded degree $d$. With an additional assumptions, we prove that the special values of partial zeta function at $s=0$ behaves as quasi-polynomial. We apply this to obtain that the special values the Hecke's $L$-functions at $s=0$ for a family of for a Dirichlet character $\chi$ behave as quasi-polynomial as well. We compute out explicitly the coefficients of the quasi-polynomials. Two examples satisfying the condition are presented and for these families the special values of the partial zeta functions at $s=0$.