Abstract
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 → ∞.
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 → ∞.
Original language  English 

Pages (fromto)  740762 
Journal  Journal of Number Theory 
Volume  238 
Early online date  26 May 2022 
Publication status  Published  1 Sept 2022 
Equipment

HPC (High Performance Computing) and HTC (High Throughput Computing) Facilities
Sadaf R Alam (Manager), Steven A Chapman (Manager), Polly E Eccleston (Other), Simon H Atack (Other) & D A G Williams (Manager)
Facility/equipment: Facility