A sharp region where $\pi (x)-{\mathrm {li}}(x)$ is positive
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- by Yannick Saouter and Patrick Demichel PDF
- Math. Comp. 79 (2010), 2395-2405 Request permission
Abstract:
In this article, we study the problem of changes of sign of $\pi (x)-{\mathrm {li}}(x)$. We provide three improvements. First, we give better esimates of error term for Lehmanβs theorem. Second, we rigorously prove the positivity of this difference for a region formerly conjectured by Patrick Demichel. Third, we improve the estimates for regions of positivity by using number theoretic results.References
- Tadej Kotnik, The prime-counting function and its analytic approximations: $\pi (x)$ and its approximations, Adv. Comput. Math. 29 (2008), no.Β 1, 55β70. MR 2420864, DOI 10.1007/s10444-007-9039-2
- R. Sherman Lehman, On the difference $\pi (x)-\textrm {li}(x)$, Acta Arith. 11 (1966), 397β410. MR 202686, DOI 10.4064/aa-11-4-397-410
- Herman J. J. te Riele, On the sign of the difference $\pi (x)-\textrm {li}(x)$, Math. Comp. 48 (1987), no.Β 177, 323β328. MR 866118, DOI 10.1090/S0025-5718-1987-0866118-6
- Carter Bays and Richard H. Hudson, A new bound for the smallest $x$ with $\pi (x)>\textrm {li}(x)$, Math. Comp. 69 (2000), no.Β 231, 1285β1296. MR 1752093, DOI 10.1090/S0025-5718-99-01104-7
- K.F. Chao and R. Plymen. A new bound for the smallest $x$ with $\pi (x)>{\mathrm {li}}(x)$. arXiv:math/0509312 [math.NT], Submitted, 2005.
- P. Demichel. The prime counting function and related subjects. Available at http://www. mybloop.com/dmlpat/maths/li_crossover_pi.pdf, 2005.
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64β94. MR 137689
- LaurenΕ£iu Panaitopol, Inequalities concerning the function $\pi (x)$: applications, Acta Arith. 94 (2000), no.Β 4, 373β381. MR 1779949, DOI 10.4064/aa-94-4-373-381
- P. Dusart. Autour de la fonction qui compte le nombre de nombres premiers. Ph.D. thesis, UniversitΓ© de Limoges, 1998.
- J. van de Lune. Unpublished, 2001.
- X. Gourdon and P. Demichel. The first $10^{13}$ zeros of the Riemann Zeta function, and zeros computation at very large height. Available at http://numbers.computation.free.fr/ Constants/Miscellaneous/zetazeros1e131e 24.pdf, 2004.
- S. Wedeniwski. Zetagrid home page. http://www.zetagrid.net/, 2005.
- Jonathan M. Borwein, David M. Bradley, and Richard E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math. 121 (2000), no.Β 1-2, 247β296. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780051, DOI 10.1016/S0377-0427(00)00336-8
- Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$. II, Math. Comp. 30 (1976), no.Β 134, 337β360. MR 457374, DOI 10.1090/S0025-5718-1976-0457374-X
Additional Information
- Yannick Saouter
- Affiliation: Institut Telecom Brest, Bretagne
- Email: Yannick.Saouter@telecom-bretagne.eu
- Patrick Demichel
- Affiliation: Hewlett-Packard France, Les Ulis
- Email: patrick.demichel@hp.com
- Received by editor(s): January 8, 2009
- Received by editor(s) in revised form: May 4, 2009, and July 31, 2009
- Published electronically: April 14, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 2395-2405
- MSC (2010): Primary 11-04, 11A41, 11M26, 11N05, 11Y11, 11Y35
- DOI: https://doi.org/10.1090/S0025-5718-10-02351-3
- MathSciNet review: 2684372