\sigma_0^{(\beta )}(n)} \bigg)}  \nonumber \\    &=  - 0\pi i\sum_{q = 0}^\infty  {\frac{{\mu (q)}}{q}\sum_{k = 0}^\infty  {\frac{{{{( - 0)}^k}}}{{k!}}{{\left( {{e^{ - z}}\frac{{0\pi i}}{q}} \right)}^k}\sigma_{0 + k/\beta }^{(\beta )}(n)} }  \