Definition and Expansions

From DRMF
Jump to navigation Jump to search

Definition and Expansions

Definition

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \sum_{n=1}^\infty \frac{1}{n^s} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}}


Other Infinite Series

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{1 - 2^{-s}} \sum_{n=0}^\infty \frac{1}{(2n+1)^s} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{1 - 2^{1-s}} \sum_{n=1}^\infty \frac{\opminus^{n-1}}{n^s} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 0}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{\opminus^n}{n!} \StieltjesConstants{n} (s-1)^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 0}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta'@{s} = - \sum_{n=2}^\infty (\ln@@{n}) n^{-s} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta^{(k)}@{s} = \opminus^k \sum_{n=2}^\infty (\ln@@{n})^k n^{-s} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k = 1,2,3,\dots}}


Representations by the Euler-Maclaurin Formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - s \int_N^\infty \frac{x-\floor{x}}{x^{s+1}} \diff{x} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 0}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle N = 1,2,3,\dots}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - \frac{1}{2}N^{-s} + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} N^{1-s-2k} - \binom{s+2n}{2n+1} \int_N^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > -2n}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n,N = 1,2,3,\dots}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \frac{1}{2} + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} - \binom{s+2n}{2n+1} \int_1^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > -2n}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n = 1,2,3,\dots}}


Infinite Products

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \prod_p (1 - p^{-s})^{-1} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}} &
product over all primes Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle p}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{(2 \cpi)^s \expe^{-s -(\EulerConstant s/2)}} {2(s-1) \EulerGamma@{\tfrac{1}{2} s + 1}} \prod_\rho \left( 1 - \frac{s}{\rho} \right) \expe^{s/\rho} }}

Constraint(s): product over zeros Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \rho}} of Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta}} with Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{\rho} > 0}}


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Definition_and_Expansions&oldid=2207"