Asymptotic Approximations

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Asymptotic Approximations

Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \RiemannZetain 2:3"): {\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): Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \iunitin 1:44"): {\displaystyle {\displaystyle s = \sigma + \iunit t}} &
Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \EulerGammain 2:1"): {\displaystyle {\displaystyle {\displaystyle \chi(s) = \pi^{s-\frac{1}{2}} \EulerGamma@{\tfrac{1}{2}-\tfrac{1}{2}s} / \EulerGamma@{\tfrac{1}{2}s}}}} &


Constraint(s): &
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formula valid as with fixed


Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \RiemannZetain 2:3"): {\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