\RiemannZeta @ σ + i t = ∑ 1 ≤ n ≤ x 1 n s + χ ( s ) ∑ 1 ≤ n ≤ y 1 n 1 - s + \BigO @ x - σ + \BigO @ y σ - 1 t 1 2 - σ \RiemannZeta @ 𝜎 imaginary-unit 𝑡 subscript 1 𝑛 𝑥 1 superscript 𝑛 𝑠 𝜒 𝑠 subscript 1 𝑛 𝑦 1 superscript 𝑛 1 𝑠 \BigO @ superscript 𝑥 𝜎 \BigO @ superscript 𝑦 𝜎 1 superscript 𝑡 1 2 𝜎 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{\sigma+\mathrm{i}t}=% \sum_{1\leq n\leq x}\frac{1}{n^{s}}+\chi(s)\sum_{1\leq n\leq y}\frac{1}{n^{1-s% }}+\BigO@{x^{-\sigma}}+\BigO@{y^{\sigma-1}t^{\frac{1}{2}-\sigma}}}}} {\displaystyle \RiemannZeta@{\sigma+\iunit t} = \sum_{1 \leq n \leq x} \frac{1}{n^s} + \chi(s) \sum_{1 \leq n \leq y} \frac{1}{n^{1-s}} + \BigO@{x^{-\sigma}} + \BigO@{y^{\sigma-1} t^{\frac{1}{2} - \sigma}} }
\RiemannZeta @ 1 2 + i t = ∑ n = 1 m 1 n 1 2 + i t + χ ( 1 2 + i t ) ∑ n = 1 m 1 n 1 2 - i t + \BigO @ t - 1 / 4 \RiemannZeta @ 1 2 imaginary-unit 𝑡 superscript subscript 𝑛 1 𝑚 1 superscript 𝑛 1 2 imaginary-unit 𝑡 𝜒 1 2 imaginary-unit 𝑡 superscript subscript 𝑛 1 𝑚 1 superscript 𝑛 1 2 imaginary-unit 𝑡 \BigO @ superscript 𝑡 1 4 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{\tfrac{1}{2}+\mathrm{i% }t}=\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}+\mathrm{i}t}}+\chi\left(\tfrac{1}{2}% +\mathrm{i}t\right)\sum_{n=1}^{m}\frac{1}{n^{\frac{1}{2}-\mathrm{i}t}}+\BigO@{% t^{-1/4}}}}} {\displaystyle \RiemannZeta@{\tfrac{1}{2}+\iunit t} = \sum_{n=1}^m \frac{1}{n^{\frac{1}{2}+\iunit t}} + \chi\left( \tfrac{1}{2}+\iunit t \right) \sum_{n=1}^m \frac{1}{n^{\frac{1}{2}-\iunit t}} + \BigO@{t^{-1/4}} }