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, a dyadic decomposition, and Abel summation, 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, we provide a theoretical complement linking integral convergence, pointwise bounds and analyticity, and conclude 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. Generally speaking, if there is an equivalence between strong and weak results for a proposition, the situation may often be doubted or considered fundamentally error, but this viewpoint is one-sided and may not necessarily be correct, because the example of a proposition involving the Riemann hypothesis that we are arguing here exactly refutes this one-sided viewpoint. In particular, we conclude that the property of absolute convergence of the integral $\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ for every $\varepsilon>0$ and the property of conditional convergence of the integral $\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ for every $\varepsilon>0$ are not only valid but also equivalent, 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 that the asymptotic relation $\psi(x)-x= o(x^{\frac{1}{2}+\varepsilon})$ for every $\varepsilon>0$ and the asymptotic relation $\psi(x)-x= O(x^{\frac{1}{2}+\varepsilon})$ for every $\varepsilon>0$ are mutually inclusive and equivalent, all of them imply the $O$-bound: $\psi(x)-x= O(x^{\frac{1}{2}+\varepsilon})$ is also valid for every $\varepsilon>0,$ which is the well-known equivalent form of the Riemann hypothesis, thus reconfirming the validity of the Riemann hypothesis.