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

5 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

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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) = 2XX (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",

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language = "English",

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journal = "Journal of Number Theory",

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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) = 2XX (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) = 2XX (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

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