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

2020/08/14 by Brent, Richard P., Platt, David J., Trudgian, Timothy S. · 1 citation
#11M06 #11M26 #11N05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2008.06140

Abstract

We show that, on the Riemann hypothesis, \limsup_X→∞I(X)/X² ≤ 0.8603, where I(X) = ∫_X^2X (ψ(x)-x)² dx. This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that (1)/(5 374)≤ I(X)/X² for sufficiently large X, and that the I(X)/X² has no limit as X→∞.

Cited by

• On Goldbach numbers in short intervals

Related

• Some explicit estimates for the error term in the prime number theorem
• On the mean values of the error terms in Mertens' theorems
• Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis
• Sharper bounds for the error term in the Prime Number Theorem
• On the sign changes of ψ(x)-x