I tried to follow the argument in the Selberg's proof of the elementary proof from the Selberg symmetry formula $$L\Lambda + \Lambda*\Lambda =L^2*\mu $$ as presented in the book The Prime Number Theorem of G.J.O. Jameson and in the review of Diamond (https://www.ams.org/journals/bull/1982-07-03/S0273-0979-1982-15057-1/), but I haven't been able to get it, specifically the part of the "recursive" bound for $R(x) = [x] — \psi(x)$ (Equation 6.7, Diamond). $$\dfrac{|R(x)|}{x}\leq \dfrac{1}{\log(x)}\int_1^x \dfrac{R(u)}{u^{2}} dx + O \left(\dfrac{\log (\log(4x))}{\log(x)}\right)$$ The estimations of the main terms of the formula are clear, but can someone help me by explaining the iterative bound on $R(x)$ part to achieve the desired bound?