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

David J Platt; Timothy Trudgian; Richard Brent

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 journal › Article (Academic Journal) › peer-review

9 Citations (Scopus)

Abstract

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 language	English
Pages (from-to)	740-762
Journal	Journal of Number Theory
Volume	238
Early online date	26 May 2022
Publication status	Published - 1 Sept 2022

Access to Document

http://10.1016/j.jnt.2021.09.016Licence: Unspecified

Handle.net

Persistent link

HPC (High Performance Computing) and HTC (High Throughput Computing) Facilities
Alam, S. R. (Manager), Williams, D. A. G. (Manager), Eccleston, P. E. (Manager) & Greene, D. (Manager)
Facility/equipment: Facility

Cite this

@article{24bc083434424d87bf5074d92f93fa38,

title = "The mean square of the error term in the prime number theorem",

abstract = "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 → ∞.",

author = "Platt, \{David J\} and Timothy Trudgian and Richard Brent",

year = "2022",

month = sep,

day = "1",

language = "English",

volume = "238",

pages = "740--762",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - The mean square of the error term in the prime number theorem

AU - Platt, David J

AU - Trudgian, Timothy

AU - Brent, Richard

PY - 2022/9/1

Y1 - 2022/9/1

N2 - 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 → ∞.

AB - 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 → ∞.

M3 - Article (Academic Journal)

SN - 0022-314X

VL - 238

SP - 740

EP - 762

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

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

Abstract

Access to Document

Handle.net

Fingerprint

Equipment

HPC (High Performance Computing) and HTC (High Throughput Computing) Facilities

Cite this