The mean square of the error term in the prime number theorem (Q2162811)

Let \(\Lambda\) be the von Mangoldt function, and \(\psi(x)=\sum_{n\leq x}\Lambda(n)\). In the paper under review the authors study the second order moment of the difference \(\psi(x)-x\), given by \[ I(X):=\int_X^{2X}\left(\psi(x)-x\right)^2 dx. \] The authors prove that, under assuming the Riemann Hypothesis (RH), for \(X\) sufficiently large one has \(1.86\times 10^{-4}\leq I(X)/X^2\leq 0.8603\). However, on the other hand, it is known that if RH is false, then \(I(X)/X^2\) is unbounded. They also prove, unconditionally, that \(\lim_{X\to\infty}I(X)/X^2\) does not exist.