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In Stein and Shakarchi's Complex Analysis, chapter 7, p.204, they give the following problem.

[Problem 4] Show that $$ \pi(x) - \text{Li}(x) = \mathcal{O}(x^{\alpha+\epsilon}) \quad \text{ as } x \to \infty $$ for every $\epsilon > 0$ (and fixed $1/2 \leq \alpha < 1$) if and only if $\zeta(z)$ has no zeros in the strip $\alpha < \text{Re} \, z < 1$.

Does there exist a proof hereof, that is accessible to someone that understands the material presented in Stein and Shakarchi, but not more?

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