Question Number 141345 by Dwaipayan Shikari last updated on 17/May/21
$$\frac{{log}\left(\zeta\left({s}\right)\right)}{{s}}=\int_{\mathrm{1}} ^{\infty} {J}\left({x}\right){x}^{−{s}−\mathrm{1}} {dx}\:\left(\:{Prove}\:{that}\right) \\ $$$${Here}\:\:{J}\left({x}\right)=\pi\left({x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\pi\left(\sqrt{{x}}\right)+\frac{\mathrm{1}}{\mathrm{3}}\pi\left(\sqrt[{\mathrm{3}}]{{x}}\right)+... \\ $$$$\pi\left({x}\right):=\boldsymbol{\mathrm{P}{rime}}\:\boldsymbol{{counting}}\:\boldsymbol{{function}} \\ $$