DETERMINANT EXPRESSION FOR THE CLASS NUMBER OF AN ABELIAN NUMBER FIELD

Abstract

Chowla's (inverse) problem is a deduction of linear independence over the rationals of circular functions at rational arguments from L.(1, x) ≠ 0, while determinant expressions for the (relative) class number of (subfields of) a cyclotomic field are referred to as the Maillet-Demyanenko determinants. In Wang, Chakraborty and Kanemitsu (to appear), Chowla's problem and Maillet-Demyanenko determinants (CPMD) in the case of Bernoulli polynomial entries (odd part) are unified as different-looking expressions of the (relative) class number on the grounds of the base change formula for periodic Dirichlet series, Dedekind determinant and the Euler product. Our aim here is to show that the genesis of the new theory of discrete Fourier transform as well as the Dedekind determinant is the characters of a finite Abelian group and its convolution map, thus revealing that CPMD boils down to analysis of the class number by group characters. We settle the case of Clausen function (log sine function) entries (even part) as an example. Other cases are similar.