Class number formulas, Kroneckerʼs limit formulas, Chowla–Selberg formulas and the generalized gamma functions

We revisit class number formulas, “s=0”-version of Kroneckerʼs limit formulas and Chowla–Selberg formulas for number fields, based on the theory at s=0 of zeta functions. The main tool in our investigation is Heckeʼs formula (Corollary to Theorem 1), which represents the zeta function of an algebraic number field K by the integral of Epsteinʼs zeta series summed over Zr. The formulas are derived without residue computations of zeta functions. Theorem 2 for class numbers is quite useful. For Chowla–Selberg formulas, the second gamma function with a character is utilized.