Cyclotomic matrices and a limit formula for \(h^-_p\) (Q2717592)

Since for \(3 \leq N \in \mathbb N\) the ``cotangent numbers'' \(c_j = i \cot (\pi j/N) = (1+ \zeta_N^j)/(1- \zeta_N^j)\) with \(j \in \mathcal R = \{j \mid 1 \leq j < N/2 \text{ and } (j,N)=1 \}\) are a \(\mathbb Q\)-basis for the imaginary part of the cyclotomic field \(\mathbb Q (\zeta_N)\), there are uniquely determined \(b_j \in \mathbb Q\) with NEWLINE\[NEWLINE2i \sin (2 \pi /N) = \zeta_N - \zeta_N^{-1} = \sum_{j \in \mathcal R} b_j c_j\;.NEWLINE\]NEWLINE Using ``cyclotomic matrices'' (i.e., orthogonality relations) and the Dirichlet series for the reciprocal \(L(s,\chi)^{-1}\) of Dirichlet's \(L\)-function, one obtains formulas for \(b_j\) as a conditionally convergent infinite series (involving Möbius's \(\mu\)-function and division values of the sine) or as a finite sum (involving the reciprocals of \(L(1, \chi)\), i.e., reciprocals of generalized Bernoulli numbers \(B_{\chi}\)), where \(\chi\) runs through the odd Dirichlet characters with conductor dividing \(N\). The main part of the paper is devoted to a thorough study of the denominator of the numbers \(b_j\), which in case of \(N=p\) a prime is intimately connected with (and under some additional conditions even equals -- up to a power of \(2\)) the minus part \(h_p^{-}\) of the class number of \(\mathbb Q (\zeta_p)\). Many numerical data illustrate these results and lead to the final conjecture that there exist only finitely many \(N\), for which some \(b_j\) vanishes.