Mathematical Applications

Distribution of Primes

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyshevPsi@{x} = \sum_{m=1}^\infty \sum_{p^m \leq x} \ln@@{p} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyshevPsi@{x} = x - \frac{\RiemannZeta'@{0}}{\RiemannZeta@{0}} - \sum_\rho \frac{x^\rho}{\rho} + \littleo@{1} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x \to \infty}} &
The sum is taken over the nontrivial 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@{s}}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyshevPsi@{x} = x + \littleo@{x} }}

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyshevPsi@{x} = x + \BigO@{x^{\frac{1}{2}+\epsilon}} }}

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

Euler Sums

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

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > -2k}} for every positive integer Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k}}

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

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > -2k}} for every positive integer Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \EulerSumH@{2} = 2 \RiemannZeta@{3} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \EulerSumH@{3} = \frac{5}{4} \RiemannZeta@{4} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \EulerSumH@{a} = \frac{a+2}{2} \RiemannZeta@{a+1} - \frac{1}{2} \sum_{r=1}^{a-2} \RiemannZeta@{r+1} \RiemannZeta@{a-r} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a = 2,3,4,\dots}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \EulerSumH@{-2a} \hiderel{=} \frac{1}{2} \RiemannZeta@{1-2a} \hiderel{=} -\frac{\BernoulliB{2a}}{4a} }}

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

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

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenEulerSumH@{s}{z} + \GenEulerSumH@{z}{s} = \RiemannZeta@{s} \RiemannZeta@{z} + \RiemannZeta@{s+z} }}

Constraint(s): both Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenEulerSumH@{s}{z}}} and Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenEulerSumH@{z}{s}}} are finite

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n \hiderel{=} 1}^\infty \left( \frac{\HarmonicNumber{n}}{n} \right)^2 = \frac{17}{4} \RiemannZeta@{4} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{r \hiderel{=} 1}^\infty \sum_{k \hiderel{=} 1}^r \frac{1}{rk(r+k)} = \frac{5}{4} \RiemannZeta@{3} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{r \hiderel{=} 1}^\infty \sum_{k \hiderel{=} 1}^r \frac{1}{r^2 (r+k)} = \frac{3}{4} \RiemannZeta@{3} }}

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