A series whose convergence is equivalent to the Riemann hypothesis

Claymath official description of the Riemann hypothesis claims the RH is true iff $\pi(x) = Li(x)+O(x^{1/2}\log x)$ so $\psi(x) = x+O(x^{1/2}\log^2 x)$ and I have been quite sloppy in that it implies the RH is true iff $$\sum_{n=2}^\infty \frac{\Lambda(n)-1}{n^{1/2}\log^{\color{red}{3+\epsilon}} n} < \infty\tag{1}$$

as with partial summation $$\sum_{n \le x} (\Lambda(n)-1) \frac{1}{n^{1/2}\log^{a} n}=\frac{\psi(x)-x}{x^{1/2}\log^a x}+\sum_{n \le x-1} (\psi(n)-n)O(\frac{1}{n^{3/2}\log^a n})\\ = O(\log^{2-a}(x))+\sum_{n \le x} O(\frac{1}{n\log^{a-2} n})$$

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The point is to show an effective explicit formula (p.28) $$\psi(x) =\sum_{n \le x} \Lambda(n)= x - \sum_{|\Im(\rho)| \le T} \frac{x^{\rho}}{\rho}+O(\frac{x\log^2 x}{T})=x - \sum_{k\le K} 2\Re(\frac{x^{\rho_k}}{\rho_k})+O(\frac{x\log^2 x}{K/\log K})$$ where $K = N(T)$ and the density of zeros gives $K \sim C T \log T,T \sim c K/\log K$,$\Im(\rho_k) \sim c k/\log k$.

In this form, under the RH, with $K = x^{1/2}$ it yields $$\psi(x) =x +O(x^{1/2}\log^{2+\delta})$$

Plotting those things indicates the series may converge very slowly with $\epsilon = 0$ and it is quite certain (under the RH) it converges with $\epsilon = 2$.