$\pi(x)-\operatorname{Li}(x)=O(x^{\alpha+\epsilon})$ and the zero-free half-plane $\Re s>\alpha$

3. \(^\ast\) Using the previous problem one can show that

\[ \pi (x) - \operatorname{Li} (x) = O (x ^ {\alpha + \epsilon}) \quad \mathrm{as} x \to \infty \]

for every \(\epsilon > 0\) , where \(\alpha\) is fixed and \(1 / 2 \le \alpha < 1\) if and only \({ \mathrm { i f ~ } } \zeta ( s )\) has no zeros in the strip \(\alpha < \mathrm { R e } ( s ) < 1\) . The case \(\alpha = 1 / 2\) corresponds to the Riemann hypothesis.