# Asymptotic Approximations

## Asymptotic Approximations

$\displaystyle {\displaystyle \RiemannZeta@{\sigma+\iunit t} = \sum_{1 \leq n \leq x} \frac{1}{n^s} + \chi(s) \sum_{1 \leq n \leq y} \frac{1}{n^{1-s}} + \BigO@{x^{-\sigma}} + \BigO@{y^{\sigma-1} t^{\frac{1}{2} - \sigma}} }$

Substitution(s): $\displaystyle {\displaystyle s = \sigma + \iunit t}$ &
$\displaystyle {\displaystyle {\displaystyle \chi(s) = \pi^{s-\frac{1}{2}} \EulerGamma@{\tfrac{1}{2}-\tfrac{1}{2}s} / \EulerGamma@{\tfrac{1}{2}s}}}$ &
$\displaystyle {\displaystyle t=2 \pi xy }$

Constraint(s): $\displaystyle {\displaystyle x \geq 1}$ &
$\displaystyle {\displaystyle y \geq 1}$ &
$\displaystyle {\displaystyle 0 \leq \sigma \leq 1}$ &
formula valid as $\displaystyle {\displaystyle t \to \infty}$ with $\displaystyle {\displaystyle \sigma}$ fixed

$\displaystyle {\displaystyle \RiemannZeta@{\tfrac{1}{2}+\iunit t} = \sum_{n=1}^m \frac{1}{n^{\frac{1}{2}+\iunit t}} + \chi\left( \tfrac{1}{2}+\iunit t \right) \sum_{n=1}^m \frac{1}{n^{\frac{1}{2}-\iunit t}} + \BigO@{t^{-1/4}} }$

Constraint(s): formula valid as $\displaystyle {\displaystyle t \to \infty}$