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

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

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

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

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

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

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

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

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

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

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

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

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

$\rm Using \,\, a \,\,value\,\, of\,\, {\color{red}\triangledown} \,\,to \,\,calculate\,\, the\,\, effect\,\, of\,\, N \,\,values \,\,approaching\,\ the\,\, true\,\, value$

$\,\displaystyle {\triangledown}= \frac{_{\sup}\,{N}-_{\inf}{N}}{_{\sup}\,{N}+_{\inf}{N}}$

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

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

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

$\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}$

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