黎曼ζ(s)函数-计算机算子▽

$\rm We ,, define ,,a ,, function ,, {\mathcal{O}(k)} ,, similar ,,to,, the,, Riemann,, Zeta,, function ,,\zeta(s)$

$$\mathcal{O(k)}=\displaystyle\lim_{n\to+\infty}\sum_{\tau=2}^n\frac1{\tau^{k}},,,,,,,,,,,\forall,s\in\mathbb{Z^*},,,s.t.,\zeta(s)-\mathcal{O}(s)=1$$

$\rm Functions ,,similar ,,to,, the,, zeta ,,function,, have,, the,, following ,,properties$

$$\displaystyle{\color{red}\lim_{m\to+\infty}\lim_{n\to+\infty}\sum_{j=2}^m\sum_{k=2}^n\frac1{j^{,k}}}=\lim_{n\to+\infty}\sum_{k=2}^n\mathcal{O}(k)=\frac34+\frac14=1$$

$\rm Some ,,properties,, of,, Riemann ,,functions ,,from,, Gaussian ,,rounding,, functions$

$$\displaystyle\left[\lim_{n\to+\infty}\sum_{s=2}^{n}\zeta(s)\right]= \left[\lim_{n\to+\infty}\sum_{k=1}^n{\mathcal{O}(2k)+\mathcal{O}(2k+1)}\right]=0$$

${\rm Consider,, functions ,,containing,, \alpha},,,,{\color{red}\displaystyle f_{\alpha}(x)=\frac1{x^{\alpha}}}$

$$\displaystyle\lim_{n\to+\infty}\sum_{x=1}^{n}f_{\alpha}(x)= 1+\frac1{2^{\alpha}}+\frac1{3^{\alpha}}+\frac1{4^{\alpha}}+\cdots$$

$\rm Precision ,,positive,, integer ,,P,, for ,,seeking,, the,, exact,, value$

$\displaystyle\forall{\alpha\gt1},$,$,\displaystyle\exists N$,$, s.t.\left|\displaystyle\lim_{n\to+\infty}\sum_{x=1}^{n}f_{{\alpha}}(x)-\sum_{x=1}^{N-1}f_{\alpha}(x)\right|=\displaystyle\lim_{n\to+\infty}\sum_{x=N}^{n}f_{\alpha}(x)\leq10^{-\rm P}$

$\rm For,, example,,,, \pi,, and ,,{\color{red}\gamma},,Euler$-$\rm Mascheroni,, constant ,,commonly,, used,, in,, number,, theory$

$$,\displaystyle \pi=3+\displaystyle\frac{1^2}{6+\displaystyle\frac{3^2}{6+\displaystyle\frac{5^2}{6+\displaystyle\frac{7^2}{6+\displaystyle\frac{9^2}{6+\ddots}}}}},\dot{=},{\tt3.14159},,,,,,,,{\pi}_{p}={\tt3.14} ,,,,,,(,\rm P=2,)$$

$,\displaystyle \gamma=\lim_{n\to+\infty}\left(\sum_{k=1}^n\frac1k-\ln n\right)\dot{=},{\tt0.577215664901532},,,,,,,,{\gamma}_{,p}={\tt0.5772156} ,,,,,,(,\rm P=7,)$

$\rm Using ,, a ,,value,, of,, {\color{red}\triangledown} ,,to ,,calculate,, the,, effect,, of,, N ,,values ,,approaching,\ the,, true,, value$
$$,\displaystyle {\triangledown}= \frac{{\sup},{N}-{\inf}{N}}{{\sup},{N}+{\inf}{N}}$$

$\rm The ,,meaning,, of ,,\triangledown,, is ,,the,, difference,, between,, the,, error,, interval,, and ,,the,, estimated,, value$

$\rm Simple ,, and ,,authentic,, series ,,scaling,, method$

$$\displaystyle\frac1{N^{\alpha-1}}{\bigg[\zeta(\alpha)-1\bigg]}\leq\lim_{n\to+\infty}\sum_{x=N}^{n}f_{\alpha}(x)\leq\frac1{N^{\alpha-1}}\left(1+\frac1{2^\alpha}+\cdots\right)=\frac{1}{N^{\alpha-1}},\zeta(\alpha)$$

$\rm Solve,, inequalities,, to,, obtain,, sets,, about,,$$N_{,\rm sim}$

$$N_{,\rm sim}=\left{x\in\mathbb Z^{*}:, \left [{\zeta(\alpha)-1}\cdot 10^{\rm P} \right]^\frac1{\alpha-1} \leq x\leq \left\lceil \left [\zeta(\alpha)\cdot 10^{\rm P} \right]^\frac1{\alpha-1} \right\rceil \right}$$

$\rm It,, seems,, that,, the ,,value ,,of,,\triangledown ,,will,, fluctuate ,,greatly,, with ,,the,, change,, of,, P$

$$\triangledown_{,\rm sim}=\displaystyle\frac{\left\lceil \left [\zeta(\alpha)\cdot 10^{\rm P} \right]^\frac1{\alpha-1} \right\rceil - \left [{\zeta(\alpha)-1}\cdot 10^{\rm P} \right]^\frac1{\alpha-1}}{\left\lceil \left [\zeta(\alpha)\cdot 10^{\rm P} \right]^\frac1{\alpha-1} \right\rceil + \left [{\zeta(\alpha)-1}\cdot 10^{\rm P} \right]^\frac1{\alpha-1}}\geq \frac{10^{\frac{P-1}{\alpha-1}}\left[10^{\frac1{\alpha-1}}-7^{\frac1{\alpha-1}}-1\right]}{3\left\lceil \left [\zeta(\alpha)\cdot 10^{\rm P} \right]^\frac1{\alpha-1} \right\rceil}$$

$\rm Approaching ,,the ,,value,, of ,,the ,,Zeta,, function,, using,, an ,,integral,, function$

        $\rm The,, Harmony,, of ,,Riemann:,,$S${,\rm rectangle}\leq ,$S$,{\zeta(s),\rm integral}\leq ,$S$_{,\rm trapezoid}$

${\forall,k ,,,,\rm integrable ,,function ,,}f_{{\alpha}}(k),,{\rm having},\displaystyle,,\displaystyle f_{{\alpha}}(k)\leq\int_{k-1}^kf_{\alpha}(x),\mathrm{d}x\leq\frac12\bigg(f_{\alpha}(k-1)+f_{\alpha}(k)\bigg)$

$$,\displaystyle\lim_{n\to+\infty}\sum_{x=N}^{n}f_{\alpha}(x)\leq\int_{N-1}^{+\infty}f_{\alpha}(x),\mathrm{d}x\leq\displaystyle\lim_{n\to+\infty}\sum_{x=N}^{n}f_{\alpha}(x)+\frac12f_{\alpha}(N-1)$$

$\rm Derive ,, the,, set,,$$N$$,,\rm from,, the ,,relationship,, between,, integrals,, and,, series$

$$N_{,\rm integral}=\left{x\in\mathbb Z^{*}: \displaystyle \frac{2x-\alpha}{\alpha-1}\cdot 10^{,p}\leq2(x-1)^\alpha , \bigcap, x \leq \left\lceil\left(\displaystyle\frac{10^{P}}{{\alpha}-1}\right)^{\frac1{\alpha-1}}+1\right\rceil\right}$$

$\rm The ,,lower ,,of,, set,,$$N$$,,\rm is,, difficult,, to,, solve,,,, resulting,, in ,,a ,,large,, deviation,, in,, \triangledown_{{\rm integral}},_.$

$\rm Function,,$$\displaystyle{\color{red}\frac{\partial,f_{\alpha}(x)}{\partial, x}}=-\alpha\cdot f_{\alpha+1}(x)\searrow$$,,,\rm it's,, known ,,that,, the,, following,, equation ,,holds$

$\rm Think ,,about,, fitting,, functions$

$$\displaystyle \left{ \begin{array}{c}
\mathcal{C}\left(\displaystyle\frac{2k-1}{2}\right)=\displaystyle\frac1{\left(\displaystyle\frac{2k-1}{2}+\vartheta\right)^\alpha}=f_\alpha(k)\
\
\mathcal{S}\left(\displaystyle k-1\right)=\displaystyle\frac1{\left(k-1+\phi\right)^\alpha}=f_{\alpha}(k)
\end{array} \right.,,\Longrightarrow,,,,,,\left{ \begin{array}{c}
\vartheta=\displaystyle\frac{1}{2}\
\
\phi=\displaystyle{1}\
\end{array} \right. $$

${\mathcal So},,\rm the,, following,, inequality,, holds,, true$

$$\displaystyle{\displaystyle\int_{k-1}^k}f_{\alpha}(x+1),\mathrm{d}x\leq{\displaystyle f_{{\alpha}}(k)\leq\int_{k-1}^k}f_{\alpha}(x+\frac12),\mathrm{d}x$$

$\rm Sum ,,up ,,functions,, and,, integrals,, from,,$$N$$\rm ,,to,, n,$$(n\to+\infty)$$\rm,,simultaneously$

$$,\displaystyle\lim_{n\to+\infty}\int_{N-1}^nf_{\alpha}(x+1),\mathrm{d}x\leq\lim_{n\to+\infty}\sum_{x=N}^{n}f_{\alpha}(x)\leq\displaystyle\lim_{n\to+\infty}\int_{N-1}^nf_{\alpha}(x+\frac12),\mathrm{d}x$$

$\rm Resolve,, the ,,inequality,, above$

$$N_{,\rm midIntegral}=\left{x\in\mathbb Z^{*}: \displaystyle \left\lceil\left(\displaystyle\frac{10^{P}}{{\alpha}-1}\right)^{\frac1{\alpha-1}}\right\rceil \leq x \leq \left\lceil\left(\displaystyle\frac{10^{P}}{{\alpha}-1}\right)^{\frac1{\alpha-1}}+\frac12\right\rceil\right}$$

$\rm The,, minimum,, value,, that,, satisfies,, the,, condition ,,is ,,equal ,,to,,$${\inf} ,N$$,,\rm or ,,$${\sup},N,$.

$$\color{red}\triangledown_{\rm midIntegral}\displaystyle\leq\frac{1}{\left(\displaystyle\frac{10^{P}}{\displaystyle{\alpha}-1}\right)^{\frac1{\alpha-1}}}$$

$\rm The ,,exact,, value,, of ,,$$N_{\rm ,precise}$

$$N_{\rm ,precise} = \left{ \displaystyle \left\lceil\left(\displaystyle\frac{10^{P}}{{\alpha}-1}\right)^{\frac1{\alpha-1}}\right\rceil,{\lor},, \displaystyle \left\lceil\left(\displaystyle\frac{10^{P}}{{\alpha}-1}\right)^{\frac1{\alpha-1}}+\frac12\right\rceil \right}$$