  We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq
0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and
improves on) a claim by Pintz from 1982. We also show unconditionally that
$\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the
$I(X)/X^{2}$ has no limit as $X\rightarrow\infty$.
