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

Abstract

We show that, on the Riemann hypothesis, lim supX→∞I(X)/X2⩽0.8603, where I(X)=∫X2X(ψ(x)−x)2dx. 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→∞. Show more