User:Misha622/sandbox

$$\lim_{m\to\infty}\sum_{n=1}^m\frac{\varphi(zn)}{m^2}=\frac{3z^2}{\psi(z)\pi^2}$$ $$ \forall z\in\mathbb{N}$$

$$\lim_{m\to\infty}\sum_{n=1}^m\frac{\sigma_k(zn)}{m^{k+1}}=\frac{\zeta(k+1)}{k+1}\prod_{i=1}^j\frac{p_i^{ke_i+1}+(p_i-1)(\frac{1-p_i^{ke_i}}{1-p_i^k})}{p_i} \forall z,k\in\mathbb{N}, z=\prod_{i=1}^j{p_i^{e_i}}$$

$$\zeta(2x+1)=(\frac{1}{x+1})(\sum_{i=1}^\infty\frac{H_i}{x^{2i}}+\sum_{j=2}^x{\zeta(j)\zeta(2x+1-j)}) \forall x\in\mathbb{N}$$

$$\mathrm{2H}_{2_g}+\mathrm{O}_{2_g}\rightarrow\mathrm{2H_2O}_{_g}$$

$$\mathrm{HCl}_{_{aq}}\rightarrow\mathrm{H}_{_{aq}}^++\mathrm{Cl}_{_{aq}}^-$$


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