Hecke's inverse cotangent numbers

Abstract

In connection with Eisenstein series for the principal congruence subgroup $\Gamma(n)$, Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the $n$th cyclotomic field or its maximal real subfield. They arise from the well investigated {\em cotangent numbers} by matrix inversion, which is why we call them {\em inverse cotangent numbers}. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that $n$ is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums.