Sharper bounds for the Chebyshev functions and . II
Author:
Lowell Schoenfeld
Journal:
Math. Comp. 30 (1976), 337360
MSC:
Primary 10H05
Corrigendum:
Math. Comp. 30 (1976), 900.
Corrigendum:
Math. Comp. 30 (1976), 900.
MathSciNet review:
0457374
Fulltext PDF Free Access
Abstract 
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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.
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 MATEST M. AGREST & MICHAIL S. MAKSIMOV, Teorija nepolnyh cilindričeskih funkciĭ i ih priloženija, Atomizdat, Moscow, 1965. Translated as Theory of incomplete cylindrical functions and their applications, Die Grundlehren der math. Wissenschaften, Band 160, SpringerVerlag, New York, Heidelberg, and Berlin, 1971. MR 32 #7796; 49 #10935. MR 0346209 (49:10935)
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 [3]
 RICHARD P. BRENT, "The first occurrence of large gaps between successive primes," Math. Comp., v. 27, 1973, pp. 959963. MR 48 #8360. MR 0330021 (48:8360)
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 RICHARD P. BRENT, "Irregularities in the distribution of primes and twin primes," Math. Comp., v. 29, 1975, pp. 4356. MR 0369287 (51:5522)
 [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, 18811886 (1884), pp. 183308.
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 HELGE VON KOCH, "Sur la distribution des nombres premiers," Acta Math., v. 24, 1901, pp. 159182. MR 1554926
 [8]
 L. J. LANDER & T. R. PARKIN, "On first appearance of prime differences," Math. Comp., v. 21, 1967, pp. 483488. MR 37 #6237. MR 0230677 (37:6237)
 [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 & LOWELL SCHOENFELD, "Approximate formulas for some functions of prime numbers," Illinois J. Math., v. 6, 1962, pp. 6494. MR 25 #1139. MR 0137689 (25:1139)
 [11]
 J. BARKLEY ROSSER & LOWELL SCHOENFELD, "Sharper bounds for the Chebyshev functions and ," Math. Comp., v. 29, 1975, pp. 243269. MR 0457373 (56:15581a)
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DOI:
http://dx.doi.org/10.1090/S0025571819760457374X
PII:
S 00255718(1976)0457374X
Article copyright:
© Copyright 1976
American Mathematical Society
