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 $m=3$, $x^3-1=(x-1)(1+x+x^2)$, leads to ${\int }_0^{1}dx\frac{1}{(1+x+x^2)}=\frac{\pi }{(3\sqrt{3})}=(1-\frac{1}{2})+(\frac{1}{4}-\frac{1}{5})+(\frac{1}{7}-\frac{1}{8})+\dots$ . 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.

This paper has been accepted in The Scientific World Journal, and will appear in brief