On the first sign change of $\theta (x) -x$
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- by D. J. Platt and T. S. Trudgian PDF
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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 $\theta (x) >x.$References
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Additional Information
- D. J. Platt
- Affiliation: Heilbronn Institute for Mathematical Research University of Bristol, Bristol, United Kingdom
- MR Author ID: 1045993
- Email: dave.platt@bris.ac.uk
- T. S. Trudgian
- Affiliation: Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia
- MR Author ID: 909247
- Email: timothy.trudgian@anu.edu.au
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 1539-1547
- MSC (2010): Primary 11M26, 11Y35
- DOI: https://doi.org/10.1090/mcom/3021
- MathSciNet review: 3454375