The mean square of the error term in the prime number theorem
| dc.contributor.author | Brent, Richard | |
| dc.contributor.author | Platt, David John | |
| dc.contributor.author | Trudgian, Tim | |
| dc.date.accessioned | 2026-01-27T23:25:32Z | |
| dc.date.available | 2026-01-27T23:25:32Z | |
| dc.date.issued | 2021 | |
| dc.date.updated | 2023-10-22T07:16:18Z | |
| dc.description.abstract | We show that, on the Riemann hypothesis, lim supX→∞I(X)/X2⩽0.8603, where I(X)=∫X2X(ψ(x)−x)2dx. 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→∞. | |
| dc.description.sponsorship | DJP is supported by Australian Research Council Discovery Project DP160100932 and EPSRC Grant EP/K034383/1 ; TST is supported by Australian Research Council Discovery Project DP160100932 and Future Fellowship FT160100094. | |
| dc.format.mimetype | application/pdf | en_AU |
| dc.identifier.issn | 0022-314X | |
| dc.identifier.uri | https://hdl.handle.net/1885/733804974 | |
| dc.language.iso | en_AU | en_AU |
| dc.publisher | Academic Press | |
| dc.relation | https://purl.org/au-research/grants/arc/DP160100932 | |
| dc.relation | https://purl.org/au-research/grants/arc/FT160100094 | |
| dc.rights | © 2021 Elsevier Inc. | |
| dc.source | Journal of Number Theory | |
| dc.title | The mean square of the error term in the prime number theorem | |
| dc.type | Journal article | |
| local.bibliographicCitation.lastpage | 762 | |
| local.bibliographicCitation.startpage | 740 | |
| local.contributor.affiliation | Brent, Richard, College of Science, ANU | |
| local.contributor.affiliation | Platt, David John, University of Bristol | |
| local.contributor.affiliation | Trudgian, Tim, UNSW Canberra | |
| local.contributor.authoruid | Brent, Richard, u4241028 | |
| local.description.embargo | 2099-12-31 | |
| local.description.notes | Imported from ARIES | |
| local.identifier.absfor | 490401 - Algebra and number theory | |
| local.identifier.absfor | 490399 - Numerical and computational mathematics not elsewhere classified | |
| local.identifier.ariespublication | a383154xPUB24022 | |
| local.identifier.citationvolume | 238 | |
| local.identifier.doi | 10.1016/j.jnt.2021.09.016 | |
| local.identifier.scopusID | 2-s2.0-85118324875 | |
| local.type.status | Published Version | |
| publicationvolume.volumeNumber | 238 |
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