The mean square of the error term in the prime number theorem

Abstract

We show that, on the Riemann hypothesis, ${\mathrm{limsup}}_{X\to \infty }I(X)/X^2\le 0.8603$, 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 $\frac{1}{5374}\le I(X)/X^2$ for sufficiently large $X$, and that the $I(X)/X^2$ has no limit as $X\to \infty$.

23 pages