The mean square of the error term in the prime number theorem
David J. Platt, Timothy S. Trudgian · Bristol Research (University of Bristol) · 2019
We show that, on the Riemann hypothesis, lim supX→∞ I(X)/X2 0.8603, where I(X) = 2X X (ψ(x) − x)2 dx. This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that 1.86 · 10−4 I(X)/X2 for sufficiently large X, and that the I(X)/X2 has no limit as X → ∞.