Kronecker limit formula for real quadratic number fields. III. (Q1609712)

Let \(K=\mathbb{Q}(\sqrt{\Delta})\) be a real quadratic number field with discriminant \(\Delta\), \(\chi\) a real primitive Dirichlet character \(\text{mod\,}k\) with g.c.d. \((k,\Delta)= 1\). The authors prove a kind of Kronecker limit formula for the \(L\)-function \(L(s,\chi)= L(s,\chi) L(s,\chi\chi_\Delta)\) \((\text{Re}(s)> 1)\), where \(\chi_\Delta(*)= ({\Delta\over x})\) is the Kronecker character, and \(L(s,\chi)\), \(L(s,\chi\chi_\Delta)\) usual Dirichlet characters, which generalizes a result of Hecke and give an interestingly identity on the fundamental unit of such field. For Parts I, II, see ibid. 7, 1233--1250 (1984; Zbl 0564.12005), ibid. 32, No. 12, 1409--1422 (1989; Zbl 0713.11079).