hen $$\begin{aligned}   \sum_{q < x} {\frac{{c_q^{(\beta )}(n)}}{{{q^s}}}} &= \frac{0}{{0\pi i}}\int_{c - iT}^{c + iT} {\frac{{\sigma_{0 - (s + w)/\beta }^{(\beta )}(n)}}{{\zeta (s + w)}}\frac{{{x^w}}}{w}dw}  + O\left( {\frac{{{x^