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

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 b n such that b −1 n = [1, δ(n)] where the continued fraction expansion of δ(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 χ behave as quasipolynomial 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.

The behavior of HeckeL-functions of real quadratic fields ats= 0

“…We conclude this section with a possible generalization of the linearity of the special value of the Hecke L-function. This generalization will be dealt in a separate paper [10].…”

“…m = qk + m q for k ∈ Z, m q ∈ [1, q] ∩ Z.). (10) [a 0 , a 1 , a 2 , ....] for positive integers a i denotes the usual continued fraction:…”

Untitled

“…We conclude with a possible generalization of the linearity of the special value of the Hecke L-function. This generalization will be dealt in [Jun and Lee 2012].…”

Higher Hickerson formula

“…where a i are terms of the continued fraction of a rational number p/q in relation with a (cf. [16], [22], [13], [15]). We are not going to give exact description of the numbers appearing here as well as the definition of the partial zeta function but refer to Sec.6 of this article.…”