Asymptotic Approximations
Asymptotic Approximations
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle s = \sigma + \iunit t}}
&
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\displaystyle \chi(s) = \pi^{s-\frac{1}{2}} \EulerGamma@{\tfrac{1}{2}-\tfrac{1}{2}s} / \EulerGamma@{\tfrac{1}{2}s}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle t=2 \pi xy }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\displaystyle \chi(s) = \pi^{s-\frac{1}{2}} \EulerGamma@{\tfrac{1}{2}-\tfrac{1}{2}s} / \EulerGamma@{\tfrac{1}{2}s}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle t=2 \pi xy }}
Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x \geq 1}}
&
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y \geq 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 0 \leq \sigma \leq 1}} &
formula valid as Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle t \to \infty}} with Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sigma}} fixed
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y \geq 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 0 \leq \sigma \leq 1}} &
formula valid as Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle t \to \infty}} with Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sigma}} fixed
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle t \to \infty}}