Relation of the Cyclotomic Equation with the Harmonic and Derived Series

Abstract EN

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 ∫ 0 1 d x ( 1 / ( 1 + x + x 2 ) ) = 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.