\[\zeta\left(\sigma+it\right)=\sum_{1\leq n\leq x}\frac{1}{n^{s}}+\chi(s)\sum_{1%
\leq n\leq y}\frac{1}{n^{1-s}}+O\left(x^{-\sigma}\right)+O\left(y^{\sigma-1}t^%
{\frac{1}{2}-\sigma}\right),\]