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 xm-1=0, for several natural numbers m; for example, for m=3, x3-1=(x-1)(1+x+x2) leads to ∫01dx(1/(1+x+x2))=π/(33)=(1-1/2)+(1/4-1/5)+(1/7-1/8)+⋯. 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.