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

David J Platt, Timothy Trudgian*, Richard Brent

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)


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 → ∞.
Original languageEnglish
Pages (from-to)740-762
JournalJournal of Number Theory
Early online date26 May 2022
Publication statusPublished - 1 Sept 2022

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