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Kronecker second limit formula for real quadratic fields

YoungJu Choie, Rahul Kumar

TL;DR

The paper derives the second Kronecker limit formula for real quadratic fields by connecting Zagier's zeta function to zeta values of narrow ideal classes through the higher Herglotz-Zagier-Novikov function $\mathscr{F}_{k}(x; \alpha, \beta)$. It establishes analytic continuation and a suite of functional equations (two-, three-, six-term) for $\mathscr{F}_{k}$, along with a cohomological interpretation via SL$_2(\mathbb{Z})$-cocycles and a link to a generalized Dedekind eta function. The main results express $D^{k/2}$-values of zeta at natural integers in terms of $\mathscr{F}_{2k}$ evaluated at reduced forms, yielding explicit two-term limits for the $k=1$ case and higher analogues. The work also provides numerical verification and identifies rational zeta values emerging from combinations of zeta values, proposing a coherent arithmetic framework with potential connections to cohomology and Stark-type phenomena.

Abstract

In this paper, the second Kronecker ``limit" formula for a real quadratic field is established for the first time. More precisely, we obtain the second Kronecker limit formula of Zagier's zeta function. Using the reduction theory of Zagier, which connects Zagier's zeta function to the zeta function of real quadratic fields, we express the values of the zeta function of narrow ideal classes in real quadratic fields at natural arguments in terms of an analytic function which we call the \emph{higher Herglotz-Zagier-Novikov function} and denote it by $\mathscr{F}_k(x; α, β)$. This function plays a central role in our study. The function $\mathscr{F}_k(x; α, β)$ possesses elegant properties, for example, we prove that it satisfies the two, three and six-term functional equations. As a result of our Kronecker limit formula and functional equations, we provide another expression for the combinations of zeta values. Finally, we interpret our Kronecker ``limit" formula in terms of cohomological relations and establish a connection between $\mathscr{F}_k(x; α, β)$ and a generalized Dedekind-eta function.

Kronecker second limit formula for real quadratic fields

TL;DR

The paper derives the second Kronecker limit formula for real quadratic fields by connecting Zagier's zeta function to zeta values of narrow ideal classes through the higher Herglotz-Zagier-Novikov function . It establishes analytic continuation and a suite of functional equations (two-, three-, six-term) for , along with a cohomological interpretation via SL-cocycles and a link to a generalized Dedekind eta function. The main results express -values of zeta at natural integers in terms of evaluated at reduced forms, yielding explicit two-term limits for the case and higher analogues. The work also provides numerical verification and identifies rational zeta values emerging from combinations of zeta values, proposing a coherent arithmetic framework with potential connections to cohomology and Stark-type phenomena.

Abstract

In this paper, the second Kronecker ``limit" formula for a real quadratic field is established for the first time. More precisely, we obtain the second Kronecker limit formula of Zagier's zeta function. Using the reduction theory of Zagier, which connects Zagier's zeta function to the zeta function of real quadratic fields, we express the values of the zeta function of narrow ideal classes in real quadratic fields at natural arguments in terms of an analytic function which we call the \emph{higher Herglotz-Zagier-Novikov function} and denote it by . This function plays a central role in our study. The function possesses elegant properties, for example, we prove that it satisfies the two, three and six-term functional equations. As a result of our Kronecker limit formula and functional equations, we provide another expression for the combinations of zeta values. Finally, we interpret our Kronecker ``limit" formula in terms of cohomological relations and establish a connection between and a generalized Dedekind-eta function.

Paper Structure

This paper contains 19 sections, 11 theorems, 170 equations.

Table of Contents

  1. Introduction and motivation
  2. The second Kronecker limit formula
  3. The second Kronecker limit formula of Zagier's zeta function
  4. The second Kronecker limit formula for the zeta function over real quadratic fields
  5. Higher Herglotz-Zagier-Novikov function
  6. Asymptotics and analytic continuation
  7. Two-, three- and six-term functional equations for $\mathscr{F}_{k}(x;\alpha,\beta)$
  8. Cohomological aspects
  9. Cocycle
  10. Connection between $\mathscr{F}_{k}(x; \alpha,\beta)$ and a generalized Dedekind eta function
  11. Numerical examples of the second Kronecker limit formula
  12. Rational zeta values
  13. Proofs
  14. Proof of asymptotics
  15. Proof of the two, three and six-term functional equations
  16. ...and 4 more sections

Key Result

Theorem 2.1

The function $Z_Q(s)$ has an analytic continuation to the half-plane $\mathrm{Re}(s)>\frac{1}{2}$ with a simple pole at $s=1$ and where $P(x, y)$ is a universal function of two variables: and where the function $F(x)$ is the Herglotz function defined in her int.

Theorems & Definitions (34)

  • Example 2.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1: The first Kronecker limit formula of Zagier's zeta function
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.2
  • Remark 3.1
  • ...and 24 more