 {\mathop {\operatorname{res} }\limits_{s = \rho } \frac{{\sigma_{0 - s/\beta }^{(\beta )}(n)}}{{\zeta (s)}}\frac{{{x^s}}}{s}}  = \sum_{|\gamma | < T} {\frac{{\sigma_{0 - \rho /\beta }^{(\beta )}(n)}}{{\zeta '(\rho )}}\frac{{{x^\rho }}}{\rho }}