title: 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 ∫(1)0 dx(1/(1 +x + x2)) = π/(3√3) = (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.
authors: Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain.
authors: Luis J
authors: LJ
authors: Boya
authors: Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain.
authors: Cristian
authors: C
authors: Rivera