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

TL;DR

This paper investigates the error term in the prime number theorem, establishing bounds on its mean square under the Riemann hypothesis and unconditionally, revealing new insights into its asymptotic behavior.

Contribution

It provides improved bounds on the mean square of the error term in the prime number theorem, both conditionally under RH and unconditionally, and shows that the ratio has no limit as X grows.

Findings

Under RH, limsup of I(X)/X^2 is at most 0.8603.

Unconditionally, I(X)/X^2 is bounded below by 1/5374 for large X.

The ratio I(X)/X^2 does not have a limit as X approaches infinity.

Abstract

We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$.