A Kronecker limit formula for totally real fields and arithmetic applications

We establish a Kronecker limit formula for the zeta function zeta(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 zeta(F)(s, A) at s = 1 to a toric integral of a SLn(Z)-invariant function log G(Z) along a Heegner cycle in the symmetric space of GL(n)(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.