Sharper bounds for the Chebyshev functions and . 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 ; 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-Heidelberg, 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****[2]**J. P. M. BINET, "Note sur l'intégrale 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**, 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**, 10.1090/S0025-5718-1975-0369287-1**[5]**HILDING FAXÉN, "Expansion in series of the integral ,"*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**, 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**, 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****[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**, 10.1090/S0025-5718-1975-0457373-7

Retrieve articles in *Mathematics of Computation*
with MSC:
10H05

Retrieve articles in all journals with MSC: 10H05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0457374-X

Article copyright:
© Copyright 1976
American Mathematical Society