Mathematical Applications

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Distribution of Primes

${\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p}}}$
${\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x-\frac{\RiemannZeta^{\prime}@{0}}{\RiemannZeta@{0}}-\sum_{\rho}\frac{x^{\rho}}{\rho}+o\left(1\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle x\to\infty}}}$ &
The sum is taken over the nontrivial zeros ${\displaystyle{\displaystyle{\displaystyle\rho}}}$ of ${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}}}}$

${\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x+o\left(x\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle x\to\infty}}}$

${\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x+\BigO@{x^{\frac{1}{2}+\epsilon}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle x\to\infty}}}$ &
${\displaystyle{\displaystyle{\displaystyle\epsilon>0}}}$

Euler Sums

${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{s}=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}}}}}$
${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{s}=-\RiemannZeta^{\prime}@{s}+\gamma\RiemannZeta@{s}+\frac{1}{2}\RiemannZeta@{s+1}+\sum_{r=1}^{k}\RiemannZeta@{1-2r}\RiemannZeta@{s+2r}+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\PeriodicBernoulliB{2k+1}@{x}}{x^{2k+2}}\mathrm{d}x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>-2k}}}$ for every positive integer ${\displaystyle{\displaystyle{\displaystyle k}}}$

${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{s}=\frac{1}{2}\RiemannZeta@{s+1}+\frac{\RiemannZeta@{s}}{s-1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\RiemannZeta@{1-2r}\RiemannZeta@{s+2r}-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}\frac{1}{n}\int_{n}^{\infty}\frac{\PeriodicBernoulliB{2k+1}@{x}}{x^{s+2k+1}}\mathrm{d}x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>-2k}}}$ for every positive integer ${\displaystyle{\displaystyle{\displaystyle k}}}$

${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{2}=2\RiemannZeta@{3}}}}$
${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{3}=\frac{5}{4}\RiemannZeta@{4}}}}$
${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{a}=\frac{a+2}{2}\RiemannZeta@{a+1}-\frac{1}{2}\sum_{r=1}^{a-2}\RiemannZeta@{r+1}\RiemannZeta@{a-r}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a=2,3,4,\dots}}}$

${\displaystyle{\displaystyle{\displaystyle\EulerSumH@{-2a}=\frac{1}{2}\RiemannZeta@{1-2a}=-\frac{\BernoulliB{2a}}{4a}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a=1,2,3,\dots}}}$

${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{n}\frac{1}{m^{z}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{(s+z)}>1}}}$

${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}+\GenEulerSumH@{z}{s}=\RiemannZeta@{s}\RiemannZeta@{z}+\RiemannZeta@{s+z}}}}$

Constraint(s): both ${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}}}}$ and ${\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{z}{s}}}}$ are finite

${\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\RiemannZeta@{4}}}}$
${\displaystyle{\displaystyle{\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}=\frac{5}{4}\RiemannZeta@{3}}}}$
${\displaystyle{\displaystyle{\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{r^{2}(r+k)}=\frac{3}{4}\RiemannZeta@{3}}}}$

<< Dirichlet L-functions

Mathematical Applications

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