Mathematical Applications

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Mathematical Applications

Distribution of Primes

ψ ⁡ ( x ) = ∑ m = 1 ∞ ∑ p m ≤ x ln ⁡ p Chebyshev-psi 𝑥 superscript subscript 𝑚 1 subscript superscript 𝑝 𝑚 𝑥 𝑝 {\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=\sum_{m=1}^{% \infty}\sum_{p^{m}\leq x}\ln p}}} {\displaystyle \ChebyshevPsi@{x} = \sum_{m=1}^\infty \sum_{p^m \leq x} \ln@@{p} }
ψ ⁡ ( x ) = x - \RiemannZeta ′ ⁢ @ ⁢ 0 \RiemannZeta ⁢ @ ⁢ 0 - ∑ ρ x ρ ρ + o ⁡ ( 1 ) Chebyshev-psi 𝑥 𝑥 superscript \RiemannZeta ′ @ 0 \RiemannZeta @ 0 subscript 𝜌 superscript 𝑥 𝜌 𝜌 little-o 1 {\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x-\frac{% \RiemannZeta^{\prime}@{0}}{\RiemannZeta@{0}}-\sum_{\rho}\frac{x^{\rho}}{\rho}+% o\left(1\right)}}} {\displaystyle \ChebyshevPsi@{x} = x - \frac{\RiemannZeta'@{0}}{\RiemannZeta@{0}} - \sum_\rho \frac{x^\rho}{\rho} + \littleo@{1} }

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


ψ ⁡ ( x ) = x + o ⁡ ( x ) Chebyshev-psi 𝑥 𝑥 little-o 𝑥 {\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x+o\left(x\right)% }}} {\displaystyle \ChebyshevPsi@{x} = x + \littleo@{x} }

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


ψ ⁡ ( x ) = x + \BigO ⁢ @ ⁢ x 1 2 + ϵ Chebyshev-psi 𝑥 𝑥 \BigO @ superscript 𝑥 1 2 italic-ϵ {\displaystyle{\displaystyle{\displaystyle\psi\left(x\right)=x+\BigO@{x^{\frac% {1}{2}+\epsilon}}}}} {\displaystyle \ChebyshevPsi@{x} = x + \BigO@{x^{\frac{1}{2}+\epsilon}} }

Constraint(s): x → ∞ → 𝑥 {\displaystyle{\displaystyle{\displaystyle x\to\infty}}} &
ϵ > 0 italic-ϵ 0 {\displaystyle{\displaystyle{\displaystyle\epsilon>0}}}


Euler Sums

\EulerSumH ⁢ @ ⁢ s = ∑ n = 1 ∞ H n n s \EulerSumH @ 𝑠 superscript subscript 𝑛 1 Harmonic-number 𝑛 superscript 𝑛 𝑠 {\displaystyle{\displaystyle{\displaystyle\EulerSumH@{s}=\sum_{n=1}^{\infty}% \frac{H_{n}}{n^{s}}}}} {\displaystyle \EulerSumH@{s} = \sum_{n=1}^\infty \frac{\HarmonicNumber{n}}{n^s} }
\EulerSumH ⁢ @ ⁢ s = - \RiemannZeta ′ ⁢ @ ⁢ s + γ ⁢ \RiemannZeta ⁢ @ ⁢ s + 1 2 ⁢ \RiemannZeta ⁢ @ ⁢ s + 1 + ∑ r = 1 k \RiemannZeta ⁢ @ ⁢ 1 - 2 ⁢ r ⁢ \RiemannZeta ⁢ @ ⁢ s + 2 ⁢ r + ∑ n = 1 ∞ 1 n s ⁢ ∫ n ∞ \PeriodicBernoulliB ⁢ 2 ⁢ k + 1 ⁢ @ ⁢ x x 2 ⁢ k + 2 ⁢ d x \EulerSumH @ 𝑠 superscript \RiemannZeta ′ @ 𝑠 \RiemannZeta @ 𝑠 1 2 \RiemannZeta @ 𝑠 1 superscript subscript 𝑟 1 𝑘 \RiemannZeta @ 1 2 𝑟 \RiemannZeta @ 𝑠 2 𝑟 superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 superscript subscript 𝑛 \PeriodicBernoulliB 2 𝑘 1 @ 𝑥 superscript 𝑥 2 𝑘 2 𝑥 {\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}}} {\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): ℜ ⁡ s > - 2 ⁢ k 𝑠 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-2k}}} for every positive integer k 𝑘 {\displaystyle{\displaystyle{\displaystyle k}}}


\EulerSumH ⁢ @ ⁢ s = 1 2 ⁢ \RiemannZeta ⁢ @ ⁢ s + 1 + \RiemannZeta ⁢ @ ⁢ s s - 1 - ∑ r = 1 k ( s + 2 ⁢ r - 2 2 ⁢ r - 1 ) ⁢ \RiemannZeta ⁢ @ ⁢ 1 - 2 ⁢ r ⁢ \RiemannZeta ⁢ @ ⁢ s + 2 ⁢ r - ( s + 2 ⁢ k 2 ⁢ k + 1 ) ⁢ ∑ n = 1 ∞ 1 n ⁢ ∫ n ∞ \PeriodicBernoulliB ⁢ 2 ⁢ k + 1 ⁢ @ ⁢ x x s + 2 ⁢ k + 1 ⁢ d x \EulerSumH @ 𝑠 1 2 \RiemannZeta @ 𝑠 1 \RiemannZeta @ 𝑠 𝑠 1 superscript subscript 𝑟 1 𝑘 binomial 𝑠 2 𝑟 2 2 𝑟 1 \RiemannZeta @ 1 2 𝑟 \RiemannZeta @ 𝑠 2 𝑟 binomial 𝑠 2 𝑘 2 𝑘 1 superscript subscript 𝑛 1 1 𝑛 superscript subscript 𝑛 \PeriodicBernoulliB 2 𝑘 1 @ 𝑥 superscript 𝑥 𝑠 2 𝑘 1 𝑥 {\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{(}{)}{0.0% pt}{}{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}}} {\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): ℜ ⁡ s > - 2 ⁢ k 𝑠 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-2k}}} for every positive integer k 𝑘 {\displaystyle{\displaystyle{\displaystyle k}}}


\EulerSumH ⁢ @ ⁢ 2 = 2 ⁢ \RiemannZeta ⁢ @ ⁢ 3 \EulerSumH @ 2 2 \RiemannZeta @ 3 {\displaystyle{\displaystyle{\displaystyle\EulerSumH@{2}=2\RiemannZeta@{3}}}} {\displaystyle \EulerSumH@{2} = 2 \RiemannZeta@{3} }
\EulerSumH ⁢ @ ⁢ 3 = 5 4 ⁢ \RiemannZeta ⁢ @ ⁢ 4 \EulerSumH @ 3 5 4 \RiemannZeta @ 4 {\displaystyle{\displaystyle{\displaystyle\EulerSumH@{3}=\frac{5}{4}% \RiemannZeta@{4}}}} {\displaystyle \EulerSumH@{3} = \frac{5}{4} \RiemannZeta@{4} }
\EulerSumH ⁢ @ ⁢ a = a + 2 2 ⁢ \RiemannZeta ⁢ @ ⁢ a + 1 - 1 2 ⁢ ∑ r = 1 a - 2 \RiemannZeta ⁢ @ ⁢ r + 1 ⁢ \RiemannZeta ⁢ @ ⁢ a - r \EulerSumH @ 𝑎 𝑎 2 2 \RiemannZeta @ 𝑎 1 1 2 superscript subscript 𝑟 1 𝑎 2 \RiemannZeta @ 𝑟 1 \RiemannZeta @ 𝑎 𝑟 {\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}}}} {\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): a = 2 , 3 , 4 , … 𝑎 2 3 4 … {\displaystyle{\displaystyle{\displaystyle a=2,3,4,\dots}}}


\EulerSumH ⁢ @ - 2 ⁢ a = 1 2 ⁢ \RiemannZeta ⁢ @ ⁢ 1 - 2 ⁢ a = - \BernoulliB ⁢ 2 ⁢ a 4 ⁢ a \EulerSumH @ 2 𝑎 1 2 \RiemannZeta @ 1 2 𝑎 \BernoulliB 2 𝑎 4 𝑎 {\displaystyle{\displaystyle{\displaystyle\EulerSumH@{-2a}=\frac{1}{2}% \RiemannZeta@{1-2a}=-\frac{\BernoulliB{2a}}{4a}}}} {\displaystyle \EulerSumH@{-2a} \hiderel{=} \frac{1}{2} \RiemannZeta@{1-2a} \hiderel{=} -\frac{\BernoulliB{2a}}{4a} }

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


\GenEulerSumH ⁢ @ ⁢ s ⁢ z = ∑ n = 1 ∞ 1 n s ⁢ ∑ m = 1 n 1 m z \GenEulerSumH @ 𝑠 𝑧 superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 superscript subscript 𝑚 1 𝑛 1 superscript 𝑚 𝑧 {\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}=\sum_{n=1}^{% \infty}\frac{1}{n^{s}}\sum_{m=1}^{n}\frac{1}{m^{z}}}}} {\displaystyle \GenEulerSumH@{s}{z} = \sum_{n=1}^\infty \frac{1}{n^s} \sum_{m=1}^n \frac{1}{m^z} }

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


\GenEulerSumH ⁢ @ ⁢ s ⁢ z + \GenEulerSumH ⁢ @ ⁢ z ⁢ s = \RiemannZeta ⁢ @ ⁢ s ⁢ \RiemannZeta ⁢ @ ⁢ z + \RiemannZeta ⁢ @ ⁢ s + z \GenEulerSumH @ 𝑠 𝑧 \GenEulerSumH @ 𝑧 𝑠 \RiemannZeta @ 𝑠 \RiemannZeta @ 𝑧 \RiemannZeta @ 𝑠 𝑧 {\displaystyle{\displaystyle{\displaystyle\GenEulerSumH@{s}{z}+\GenEulerSumH@{% z}{s}=\RiemannZeta@{s}\RiemannZeta@{z}+\RiemannZeta@{s+z}}}} {\displaystyle \GenEulerSumH@{s}{z} + \GenEulerSumH@{z}{s} = \RiemannZeta@{s} \RiemannZeta@{z} + \RiemannZeta@{s+z} }

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


∑ n = 1 ∞ ( H n n ) 2 = 17 4 ⁢ \RiemannZeta ⁢ @ ⁢ 4 superscript subscript 𝑛 1 superscript Harmonic-number 𝑛 𝑛 2 17 4 \RiemannZeta @ 4 {\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\left(\frac{H_{n}% }{n}\right)^{2}=\frac{17}{4}\RiemannZeta@{4}}}} {\displaystyle \sum_{n \hiderel{=} 1}^\infty \left( \frac{\HarmonicNumber{n}}{n} \right)^2 = \frac{17}{4} \RiemannZeta@{4} }
∑ r = 1 ∞ ∑ k = 1 r 1 r ⁢ k ⁢ ( r + k ) = 5 4 ⁢ \RiemannZeta ⁢ @ ⁢ 3 superscript subscript 𝑟 1 superscript subscript 𝑘 1 𝑟 1 𝑟 𝑘 𝑟 𝑘 5 4 \RiemannZeta @ 3 {\displaystyle{\displaystyle{\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}% \frac{1}{rk(r+k)}=\frac{5}{4}\RiemannZeta@{3}}}} {\displaystyle \sum_{r \hiderel{=} 1}^\infty \sum_{k \hiderel{=} 1}^r \frac{1}{rk(r+k)} = \frac{5}{4} \RiemannZeta@{3} }
∑ r = 1 ∞ ∑ k = 1 r 1 r 2 ⁢ ( r + k ) = 3 4 ⁢ \RiemannZeta ⁢ @ ⁢ 3 superscript subscript 𝑟 1 superscript subscript 𝑘 1 𝑟 1 superscript 𝑟 2 𝑟 𝑘 3 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}}}} {\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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