Riemann hypothesis equivalence for the second Chebyshev function

The Riemann hypothesis is equivalent tofor every $ϵ>0$. In fact, the hypothesis and the Riemann–von Mangoldt explicit formula give the stronger $O(x^{1/2}(\log x{)}^2)$ bound. Conversely, the stated error continues $-{\zeta }^{\prime }/\zeta -s/(s-1)$ holomorphically to $\Re s>1/2$, excluding zeros there; the functional equation supplies the other half.