Averaging effects on irregularities in the distribution of primes in arithmetic progressions

Mathematics of Computation

Volumn 44, Issue 170, 1985, Pages 561-571

  Hudson, Richard H ^a  

^a UNIVERSITY OF SOUTH CAROLINA   (United States)

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[No Author keywords available]

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EID: 84966211722     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-1985-0777286-7     Document Type: Article

References (32)

1. R. Ayoub, An Introduction to the Analytic Theory of Numbers, Math. Surveys, Amer. Math. Soc. Providence, R. I, 10, 1963.
2. Carter Bays, Richard H. Hudson, "The segmented sieve of Eratosthenes and primes in arithmetic progressions to 1012, " BIT, v. 17, 1977, p. 121-127.
3. Carter Bays, Richard H. Hudson, "On the fluctuations of Littlewood for primes of the form 4/r1." Math. Comp., v. 32, 1978, p. 281-286.
4. Carter Bays, Richard H. Hudson, "Details of the first region of integers x with w32(x) w3(x), " Math. Comp., v. 32, 1978, p. 571-576.
5. A. Bloch, G. Polya, "On the roots of certain algebraic equations, " Proc. London Math. Soc. (2), v. 33, 1932, p. 102-114.
6. Richard P. Brent, "Irregularities in the distribution of primes and twin primes, " Math. Comp., v. 29, 1975, p. 43-56.
7. Richard P. Brent, Tables Concerning Irregularities in the Distribution of Primes and Twin Primes to 10, Computer Centre, Australian National University, Canberra, Australia, 1975.
8. P. L. Chebyshev, "Lettre de M le professeur Tchebychev a M. Fuss sur un nouveau theorem relatif aux nombres premiers contenus dans les formes An + 1 et An + 3, " Bull. Classe Phys. Mathem. Acad. Imp. des Sciences, St. Petersburg, v. 11, 1853, p. 208
9. W. J. Ellison, Les Nombres Premiers, Hermann, Paris, 1975.
10. Emil Grosswald, "Oscillation theorems of arithmetic functions, " Trans. Amer. Math. Soc, v. 126, 1967, p. 1-28.
11. G. H. Hardy, J. E. Littlewood, "Contributions to the theory of the Riemann zeta function and the theory of the distribution of primes, " Ada Math., v. 41, 1917, p. 119-196.
12. Richard H. Hudson, Alfred Brauer, "On the exact number of primes in the arithmetic progressions An 1 and 6+ 1, " J. Reine Angew. Math., v. 291, 1977, p. 23-29.
13. Richard H. Hudson, "A formula for the exact number of primes below a given bound in any arithmetic progression, " Bull. Austral. Math. Soc, v. 16, 1977, pp. 67-73.
14. Richard H. Hudson, Carter Bays, "The mean behavior of primes in arithmetic progressions, " J. Reine Angew. Math., v. 296, 1977. p. 80-99
15. Richard H. Hudson, "A common combinatorial principle underlies Riemann's formula, the Chebyshev phenomenon, and other subtle effects in comparative prime number theory. I, " J. Reine Angew. Math., v. 313, 1980, p. 133-150.
16. A. E. Ingham, The Distribution of Prime Numbers, Stechert-Hafner, New York, 1964.
17. S. Knapowski, P. Turan, "Further developments in the comparative prime-number theory I, " Acta. Arith., v. 9, 1964, p. 23-40.
18. S. Knapowski, P. Turan, "Further developments in the comparative prime-number theory II, A modification of Chebyshev's assertion, " Ada Arith., v. 10, 1964, p. 293-313.
19. S. Knapowski, P. Turan, "Further developments in the comparative prime-number theory III, " Acta Arith. v. 11, 1965. p. 115-127.
20. S. Knapowski, P. Turan, "Further developments in the comparative prime-number theory IV, " Acta Arith. v. 11, 1965, p. 147-161.
21. S. Knapowski, P. Turan. "Further developments in the comparative prime-number theory V, " Ada Arith. v. 11, 1965, p. 193-202.
22. S. Knapowski, P. Turan, "Further developments in the comparative prime-number theory VI. Accumulation theorems for residue-classes representing quadratic residues mod k." Ada Arith., v. 12, 1966. p. 85-96.
23. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, New York, 1953.
24. E. Landau, "Uber einige ltere Vermutungen und Behauptungen in der Primzahltheorie, " Math. z., v. 1, 1919, p. 1-24.
25. J. E. Littlewood, "Sur la distribution des nombres premiers, " C. R. Acad. Sci. Paris, v. 158, 1914, p. 1969-1972.
26. Karl K. Norton, "Upper bounds for A-th power coset representatives modulo n, " Ada Arith. v. 15, 1968, p. 161-179.
27. G POLYA, Uber das Vorzeichen des Restgliedes im Primzahlsatz, Gottingen Nachrichten, 1930, p. 19-27.
28. B. Riemann, lieber die Anzahl der Primzahlen unter einer gegebenen Grosse, Monatsbericht. Preuss. Akad. Wissens. 1880, p. 671-680.
29. Daniel Shanks, "Quadratic residues and the distribution of primes, " Math. Comp. v. 13, 1959, p. 272-284.
30. Daniel Shanks, "Systematic examination of Littlewoods bounds on 1, (1, x), " Proc Sympos. Pure Math., vol. 24, 1973, p. 267-283.
31. Daniel Shanks, "Bateman's constants reconsidered and the distribution of cubic residues, " Math. Comp., v. 26, 1972, p. 265-285.
32. H. M. Stark, "A problem in comparative prime number theory, " Acta Arith. v. 18. 1971, p. 311-320.