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Mathematics of Computation

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On the first sign change of $ \theta(x) -x$


Authors: D. J. Platt and T. S. Trudgian
Journal: Math. Comp. 85 (2016), 1539-1547
MSC (2010): Primary 11M26, 11Y35
DOI: https://doi.org/10.1090/mcom/3021
Published electronically: August 21, 2015
MathSciNet review: 3454375
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Abstract: Let $ \theta (x) = \sum _{p\leq x} \log p$. We show that $ \theta (x)<x$ for $ 2<x< 1.39\cdot 10^{17}$. We also show that there is an $ x<\exp (727.951332668)$ for which $ \theta (x) >x.$


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Additional Information

D. J. Platt
Affiliation: Heilbronn Institute for Mathematical Research University of Bristol, Bristol, United Kingdom
Email: dave.platt@bris.ac.uk

T. S. Trudgian
Affiliation: Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia
Email: timothy.trudgian@anu.edu.au

DOI: https://doi.org/10.1090/mcom/3021
Keywords: Sign changes of arithmetical functions, oscillation theorems
Received by editor(s): July 7, 2014
Received by editor(s) in revised form: November 19, 2014
Published electronically: August 21, 2015
Additional Notes: The second author was supported by Australian Research Council DECRA Grant DE120100173
Article copyright: © Copyright 2015 American Mathematical Society