Relation of the cyclotomic equation with the harmonic and derived series

TL;DR

This paper explores the connection between cyclotomic equations and classical series, expressing integrals and series in terms of roots of unity and Dirichlet characters, revealing new relationships in number theory.

Contribution

It establishes a novel link between cyclotomic equations and harmonic series, providing explicit integral and series representations involving roots of unity and Dirichlet characters.

Findings

Derived series representations for specific cyclotomic equations

Expressed integrals in terms of Dirichlet characters

Generalized relationships for arbitrary m

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 $\int_0^1dx\frac{1}{(1+x+x^2)} = \frac{\pi}{(3\sqrt{3})} = (1-\frac{1}{2}) + (\frac{1}{4}-\frac{1}{5}) + (\frac{1}{7}-\frac{1}{8}) + \ldots$ . 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.