\[\zeta\left(\tfrac{1}{2}+it\right)=\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}+it}}+%
\chi\left(\tfrac{1}{2}+it\right)\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}-it}}+O%
\left(t^{-1/4}\right).\]