Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥). II

Schoenfeld, Lowell

doi:10.1090/S0025-5718-1976-0457374-X

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6842(online) ISSN 0025-5718(print)

Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$ . II

Author: Lowell Schoenfeld
Journal: Math. Comp. 30 (1976), 337-360
MSC: Primary 10H05
Corrigendum: Math. Comp. 30 (1976), 900.
Corrigendum: Math. Comp. 30 (1976), 900.
MathSciNet review: 0457374
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, bounds given in the first part of the paper are strengthened. In addition, it is shown that the interval contains a prime for all $x \geqslant 2,010,760$ ; and explicit bounds for the Chebyshev functions are given under the assumption of the Riemann hypothesis.

[1] M. M. Agrest and M. S. Maksimov, Theory of incomplete cylindrical functions and their applications, Springer-Verlag, New York, 1971. Translated from the Russian by H. E. Fettis, J. W. Goresh and D. A. Lee; Die Grundlehren der mathematischen Wissenschaften, Band 160. MR 0346209 (49 #10935)
[2] J. P. M. BINET, "Note sur l'intégrale $\smallint _a^y{y^{2i}}dy\;{e^{ - p/{y^2} - q{y^2}}}$ prise entre des limites arbitraires," C. R. Acad. Sci. Paris, v. 12, 1841, pp. 958-962.
[3] Richard P. Brent, The first occurrence of large gaps between successive primes, Math. Comp. 27 (1973), 959–963. MR 0330021 (48 #8360), http://dx.doi.org/10.1090/S0025-5718-1973-0330021-0
[4] Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29 (1975), 43–56. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR 0369287 (51 #5522), http://dx.doi.org/10.1090/S0025-5718-1975-0369287-1
[5] HILDING FAXÉN, "Expansion in series of the integral $\smallint _y^\infty \;{e^{ - x(t \pm {t^{ - \mu }})}}{t^v}dt$ ," Ark. Mat. Astronom. Fys., v. 15, no. 13, 1921, 57 pp.
[6] J. P. GRAM, "Undersøgelser angaaende Maengen af Primtal under en given Graense," K. Danske Vidensk. Selskabs Skrifter, Naturv. og Math. Afd. ser. 6, v. 2, 1881-1886 (1884), pp. 183-308.
[7] Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1901), no. 1, 159–182 (French). MR 1554926, http://dx.doi.org/10.1007/BF02403071
[8] L. J. Lander and T. R. Parkin, On first appearance of prime differences, Math. Comp. 21 (1967), 483–488. MR 0230677 (37 #6237), http://dx.doi.org/10.1090/S0025-5718-1967-0230677-4
[9] DERRICK NORMAN LEHMER, List of Prime Numbers from 1 to 10,006,721, Carnegie Institution of Washington, Publication No. 165, Washington, D.C., 1914; reprinted, Hafner Publishing Co., New York, 1956.
[10] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689 (25 #1139)
[11] J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥), Math. Comp. 29 (1975), 243–269. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR 0457373 (56 #15581a), http://dx.doi.org/10.1090/S0025-5718-1975-0457373-7

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|