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

Richard P. Brent; David J. Platt; Timothy S. Trudgian

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arXiv · 2008.06140

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

Richard P. Brent·David J. Platt·Timothy S. Trudgian

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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 $X\rightarrow\infty$.

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BibTeX RIS

Richard P. Brent, David J. Platt, Timothy S. Trudgian. 2020-08-14. The mean square of the error term in the prime number theorem. https://arxiv.org/abs/2008.06140

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