Riemann hypothesis equivalence for the second Chebyshev function
= Riemann hypothesis equivalence for the second Chebyshev function
{c}
The <Riemann hypothesis> is equivalent to
$$
\psi(x)=x+O_\epsilon(x^{1/2+\epsilon})
$$
for every $\epsilon>0$. 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 $-\zeta'/\zeta-s/(s-1)$ holomorphically to $\Re s>1/2$, excluding zeros there; the functional equation supplies the other half.