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Boya","submitted_at":"2015-01-22T10:54:48Z","abstract_excerpt":"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. 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