On the sign of the difference $\pi (x)-\textrm {li}(x)$
HTML articles powered by AMS MathViewer
- by Herman J. J. te Riele PDF
- Math. Comp. 48 (1987), 323-328 Request permission
Abstract:
Following a method of Sherman Lehman we show that between $6.62 \times {10^{370}}$ and $6.69 \times {10^{370}}$ there are more than ${10^{180}}$ successive integers x for which $\pi (x) - {\text {li}}(x) > 0$. This brings down Sherman Lehman’s bound on the smallest number x for which $\pi (x) - {\text {li}}(x) > 0$, namely from $1.65 \times {10^{1165}}$ to $6.69 \times {10^{370}}$. Our result is based on the knowledge of the truth of the Riemann hypothesis for the complex zeros $\beta + i\gamma$ of the Riemann zeta function which satisfy $|\gamma | < 450,000$, and on the knowledge of the first 15,000 complex zeros to about 28 digits and the next 35,000 to about 14 digits.References
- Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, DOI 10.1090/S0025-5718-1979-0537983-2 W. Gabcke, Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel, Dissertation, Universität Göttingen, 1979.
- J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing $\pi (x)$: the Meissel-Lehmer method, Math. Comp. 44 (1985), no. 170, 537–560. MR 777285, DOI 10.1090/S0025-5718-1985-0777285-5
- Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed; With an appendix by Paul T. Bateman. MR 0068565 J. E. Littlewood, "Sur la distribution des nombres premiers," C. R. Acad. Sci. Paris, v. 158, 1914, pp. 1869-1872. J. van de Lune, H. J. J. te Riele & D. T. Winter, Rigorous High Speed Separation of Zeros of Riemann’s Zeta Function, Report NW 113/81, Mathematical Centre, Amsterdam, October 1981.
- J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), no. 174, 667–681. MR 829637, DOI 10.1090/S0025-5718-1986-0829637-3 H. J. J. te Riele, Tables of the First 15,000 Zeros of the Riemann Zeta Function to 28 Significant Digits, and Related Quantities, Report NW 67/79, Mathematical Centre, Amsterdam, June 1979.
- J. Barkley Rosser, J. M. Yohe, and Lowell Schoenfeld, Rigorous computation and the zeros of the Riemann zeta-function. (With discussion), Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 70–76. MR 0258245
- R. Sherman Lehman, On the difference $\pi (x)-\textrm {li}(x)$, Acta Arith. 11 (1966), 397–410. MR 202686, DOI 10.4064/aa-11-4-397-410
- S. Skewes, On the difference $\pi (x)-\textrm {li}\,x$. II, Proc. London Math. Soc. (3) 5 (1955), 48–70. MR 67145, DOI 10.1112/plms/s3-5.1.48
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 323-328
- MSC: Primary 11Y35; Secondary 11M06
- DOI: https://doi.org/10.1090/S0025-5718-1987-0866118-6
- MathSciNet review: 866118