On a generalization of the Maillet determinant II

Abstract

Dedicated to Professor Katsuya Miyake on his sixtieth birthday 1. Introduction. In the previous paper [16] (referred to as Part I hereafter) we have rendered it visible that the two intimately connected problems of number theory, Maillet determinants and Chowla's problem, were dealt with in complete separation, save for Girstmair [11], where this interaction was duly noticed and proved to be crucial.

Determinant Expression for the Class Number of an Abelian Number Field

“…We refer to part of the definition of the Maillet determinant [8]. In the extensive list of Yamamura [15] on Maillet determinants, that in [8] is referred to as one centering around the even part. Ref.…”

“…Correspondingly to passages surrounding [7,Lemma 2.4] we state the even case in which the procedure is almost verbatim to the odd case with B k replaced by A k ; cf. [8,14,21].…”

“…Along with (generalized) Bernoulli numbers we define the value of A k (0) (the kth Clausen number, so to say) as in [8] by…”

A generalization of Newman’s formula

“…(See Girstmair [1], Hirabayashi [6] and [4].) For general types of determinant formulas see Hirabayashi [5], Kanemitsu and Kuzumaki [7] and Kučera [8].…”

Maillet determinants for real abelian number fields and its applications

“…In the present paper, for a real abelian number field K, we define an analogue Q(K) of the Maillet determinant in terms of the Clausen function in [6]. We give a class number formula for K in Section 3.…”