Relation of the cyclotomic equation with the harmonic and derived series

Abstract: We associate some (old) convergent series related to definite integrals with the cyclotomic equation $x^m-1= 0$, for several natural numbers $m$; for example, for $m = 3$, $x^3-1 = (x-1)(1+x+x<sup>2)$, leads to $$\int_0^{1dx\frac{1}{(1+x+x^2)}} = \frac{\pi}{(3\sqrt{3})} = (1-\frac{1}{2}) + (\frac{1}{4}-\frac{1}{5}) + (\frac{1}{7}-\frac{1}{8}) + \ldots$$ . In some cases, we express the results in terms of the Dirichlet characters. Generalizations for arbitrary $m$ are well defined, but do imply integrals and/or series summations rather involved.