\RiemannZeta ⢠@ â¢ Ï + i ⢠t = â 1 ⤠n ⤠x 1 n s + Ï â¢ ( s ) ⢠â 1 ⤠n ⤠y 1 n 1 - s + \BigO ⢠@ ⢠x - Ï + \BigO ⢠@ ⢠y Ï - 1 ⢠t 1 2 - Ï \RiemannZeta @ ð imaginary-unit ð¡ subscript 1 ð ð¥ 1 superscript ð ð ð ð subscript 1 ð ð¦ 1 superscript ð 1 ð \BigO @ superscript ð¥ ð \BigO @ superscript ð¦ ð 1 superscript ð¡ 1 2 ð {\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 = â n = 1 m 1 n 1 2 + i ⢠t + Ï â¢ ( 1 2 + i ⢠t ) ⢠â n = 1 m 1 n 1 2 - i ⢠t + \BigO ⢠@ ⢠t - 1 / 4 \RiemannZeta @ 1 2 imaginary-unit ð¡ superscript subscript ð 1 ð 1 superscript ð 1 2 imaginary-unit ð¡ ð 1 2 imaginary-unit ð¡ superscript subscript ð 1 ð 1 superscript ð 1 2 imaginary-unit ð¡ \BigO @ superscript ð¡ 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}} }