Proof of the Riemann Hypothesis via the Chebyshev Function and the Integral Convergence

In this article, we offer a complete, self-contained, and entirely elementary proof of the mean-square estimate for the Chebyshev function. From this we deduce the convergence of the integral is valid, thus proving the validity of the Riemann hypothesis. The proof primarily employs elementary estimates of the Chebyshev function, the Cauchy-Schwarz inequality, and a dyadic decomposition (with Abel summation applied in the appendix), in which the argument results of this article are already optimal within the elementary framework and sufficient to derive the convergence of the required integral that it is a suffficient condition for the Riemann hypothesis. In particular, the appendix of this paper provides a theoretical complement linking integral convergence, pointwise bounds and analyticity, and concludes that the well-known $o$-bound is valid, thereby reconfirming the validity of the Riemann hypothesis. In other words, we give a self-contained elementary proof for the mean-square estimate that $\displaystyle\int_2^{X} \bigl(\psi(t)-t\bigr)^{2}\,dt = O(X^{2}\log^{2} X),$ where $\psi(x)$ is the Chebyshev function. From this we deduce that $\displaystyle\int_{1}^{\infty}\frac{|\psi(x)-x|}{x^{\frac{3}{2}+\varepsilon}}\,dx < \infty$ holds for every $\varepsilon>0,$ thus concluding the integral $\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ converges absolutely for every $\varepsilon>0,$ so that the integral $\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ converges conditionally for every $\varepsilon>0,$ whereas the integral converges conditionally $\iff \text{RH},$ so then the Riemann hypothesis is true. In particular, we conclude that the property of absolute convergence of the integral $\displaystyle\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ for every $\varepsilon>0$ is equivalent to the property of conditional convergence of the integral $\displaystyle\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ for every $\varepsilon>0,$ either of which is equivalent to the property of the $o$-bound: $|\psi(x)-x| = o(x^{\frac{1}{2}+\varepsilon})$ for every $\varepsilon>0,$ and all of them imply the $O$-bound: $\psi(x)-x= O(x^{\frac{1}{2}+\varepsilon})$ is also valid for every $\varepsilon>0,$ thus reconfirming the validity of the Riemann hypothesis.