An adelic setting of some functions analogous to \(\log|\eta (z)|)\) (Q1071799)

Let \(F\) be a number field, \(\mathbb A\) the adeles of \(F\), and \(D\) an irreducible representation of the standard maximal compact subgroup of \(\mathrm{SL}(2,\mathbb A)\). The author studies the adelic Eisenstein series attached to \(D\) when (1) \(F=\mathbb Q\) and \(D_{\infty}=1\) or (2) \(F\) has class number one and \(D\) is trivial. In (1) he recovers the known Fourier expansion of classical Eisenstein series of arbitrary level. In (2) he recovers a generalization of Kronecker's first limit formula, due to \textit{T. Asai} [Nagoya Math. J. 40, 193--211 (1970; Zbl 0213.05701)].