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

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

doi:10.1016/j.jnt.2021.09.016

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

Richard P. Brent, David J. Platt, Timothy S. Trudgian^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We show that, on the Riemann hypothesis, lim sup_X→∞I(X)/X²⩽0.8603, where I(X)=∫_X^2X(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→∞.

Original language	English
Pages (from-to)	740-762
Number of pages	23
Journal	Journal of Number Theory
Volume	238
DOIs	https://doi.org/10.1016/j.jnt.2021.09.016
Publication status	Published - Sept 2022

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10.1016/j.jnt.2021.09.016

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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)=∫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→∞.",

keywords = "Distribution of primes, Limiting distribution, Prime number theory, Riemann hypothesis, Zeta function",

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

note = "{\textcopyright} 2021 Elsevier Inc.",

year = "2022",

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doi = "10.1016/j.jnt.2021.09.016",

language = "English",

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

AU - Brent, Richard P.

AU - Platt, David J.

AU - Trudgian, Timothy S.

PY - 2022/9

Y1 - 2022/9

N2 - 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→∞.

AB - 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→∞.

KW - Distribution of primes

KW - Limiting distribution

KW - Prime number theory

KW - Riemann hypothesis

KW - Zeta function

UR - https://www.scopus.com/pages/publications/85118324875

U2 - 10.1016/j.jnt.2021.09.016

DO - 10.1016/j.jnt.2021.09.016

M3 - Article

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

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