On the first sign change of θ(x)-x

David J Platt; Tim Trudgian

doi:10.1090/mcom/3021

On the first sign change of θ(x)-x

David J Platt, Tim Trudgian

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

Let θ(x) = ∑_p≤xlog_p. We show that θ(x) < x for 2 < x < 1.39·1017. We also show that there is an x < exp(727.951332668) for which θ(x) > x.

Original language	English
Pages (from-to)	1539-1547
Number of pages	9
Journal	Mathematics of Computation
Volume	85
DOIs	https://doi.org/10.1090/mcom/3021
Publication status	Published - 21 Aug 2015

Keywords

Sign changes of arithmetical functions
oscillation theorems

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10.1090/mcom/3021

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First published in Mathematics of Computation in 85 (2016), published by the American Mathematical Society.
Accepted author manuscript, 363 KB

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title = "On the first sign change of θ(x)-x",

abstract = "Let θ(x) = ∑p≤xlogp. We show that θ(x) < x for 2 < x < 1.39·1017. We also show that there is an x < exp(727.951332668) for which θ(x) > x. ",

keywords = "Sign changes of arithmetical functions, oscillation theorems",

author = "Platt, {David J} and Tim Trudgian",

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T1 - On the first sign change of θ(x)-x

AU - Platt, David J

AU - Trudgian, Tim

PY - 2015/8/21

Y1 - 2015/8/21

N2 - Let θ(x) = ∑p≤xlogp. We show that θ(x) < x for 2 < x < 1.39·1017. We also show that there is an x < exp(727.951332668) for which θ(x) > x.

AB - Let θ(x) = ∑p≤xlogp. We show that θ(x) < x for 2 < x < 1.39·1017. We also show that there is an x < exp(727.951332668) for which θ(x) > x.

KW - Sign changes of arithmetical functions

KW - oscillation theorems

U2 - 10.1090/mcom/3021

DO - 10.1090/mcom/3021

M3 - Article (Academic Journal)

VL - 85

SP - 1539

EP - 1547

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

ER -

On the first sign change of θ(x)-x

Abstract

Keywords

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HPC (High Performance Computing) Facility

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