# A Kronecker limit formula for totally real fields and arithmetic applications

- Sheng-Chi Liu
^{1}and - Riad Masri
^{2}Email author

**1**:8

**DOI: **10.1007/s40993-015-0009-3

© The Author(s) 2015

**Received: **21 April 2015

**Accepted: **26 April 2015

**Published: **1 September 2015

## Abstract

We establish a Kronecker limit formula for the zeta function *ζ*
_{
F
}(*s*,*A*) of a wide ideal class *A* of a totally real number field *F* of degree *n*. This formula relates the constant term in the Laurent expansion of *ζ*
_{
F
}(*s*,*A*) at *s*=1 to a toric integral of a \({SL}_{n}({\mathbb {Z}})\)-invariant function log*G*(*Z*) along a Heegner cycle in the symmetric space of \({GL}_{n}({\mathbb {R}})\). We give several applications of this formula to algebraic number theory, including a relative class number formula for *H*/*F* where *H* is the Hilbert class field of *F*, and an analog of Kronecker’s solution of Pell’s equation for totally real multiquadratic fields. We also use a well-known conjecture from transcendence theory on algebraic independence of logarithms of algebraic numbers to study the transcendence of the toric integral of log*G*(*Z*). Explicit examples are given for each of these results.