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)

15 Citations (Scopus)

204 Downloads (Pure)

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

Access to Document

10.1090/mcom/3021

Skewestheta8
First published in Mathematics of Computation in 85 (2016), published by the American Mathematical Society.
Accepted author manuscript, 363 KB

Fingerprint Dive into the research topics of 'On the first sign change of <i>θ(x)</i>-<i>x</i>'. Together they form a unique fingerprint.

Cite this

Platt, D. J., & Trudgian, T. (2015). On the first sign change of θ(x)-x. Mathematics of Computation, 85, 1539-1547. https://doi.org/10.1090/mcom/3021

@article{ef81d37624d340b5a064d1b8f630d199,

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",

year = "2015",

month = aug,

day = "21",

doi = "10.1090/mcom/3021",

language = "English",

volume = "85",

pages = "1539--1547",

journal = "Mathematics of Computation",

issn = "0025-5718",

publisher = "American Mathematical Society",

}

TY - JOUR

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 -