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How are the Prime Numbers Distributed?

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The Book of Prime Number Records

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Abstract

As I have already stressed, the various proofs of existence of infinitely many primes are not constructive and do not give an indication of how to determine the nth prime number. Equivalently, the proofs do not indicate how many primes are less than any given number N.

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References

  1. CAUCHY, A.L. Demonstration générale du théorème de Fermat sur les nombres polygônes. Bull. Soc. Philomatique, 1815, 196–197. Reprinted in Oeuvres, (2), Vol. 2, 202–204. Gauthier-Villars, Paris, 1903.

    Google Scholar 

  2. DIRICHLET, G.L. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abh. d. Königl. Akad. d. Wiss., 1837, 45–81. Reprinted in Werke, Vol. I, 315–350. G. Reimer, Berlin, 1889.

    Google Scholar 

  3. MEISSEL, E.D.F. Berechnung der Menge von Primzahlen, welche innerhalb der ersten Milliarde natürlicher Zahlen vorkommen. Math. Ann., 25, 1885, 251–257.

    MathSciNet  Google Scholar 

  4. SYLVESTER, J.J. On arithmetical series. Messenger of Math., 21, 1892, 1–19 and 87–120. Reprinted in Gesammelte Abhandlungen, Vol. III, 573–587. Springer-Verlag, New York, 1968.

    Google Scholar 

  5. WENDT, E. Elementarer Beweis des Satzes dass in jeder unbegrenzter arithmetischen Progression my + 1 unendlich viele Primzahlen vorkommen. Journal f. d. reine u. angew. Math., 115, 1895, 85–88.

    MATH  Google Scholar 

  6. VON KOCH, H. Sur la distribution des nombres premiers. Acta Math., 24, 1901, 159–182.

    MathSciNet  Google Scholar 

  7. WOLFSKEHL, P. Ueber eine Aufgabe der elementaren Arithmetik. Math. Ann., 54, 1901, 503–504.

    MathSciNet  MATH  Google Scholar 

  8. TORELLI, G. Sulla totalità dei numeri primi fino a un limite assegnato. Atti d. Reale Acad. d. Sci. Fis. e Mat. di Napoli, (2), 11, 1902, 1–222.

    Google Scholar 

  9. LANDAU, E. Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Math. Ann., 56, 1903, 645–670.

    MathSciNet  MATH  Google Scholar 

  10. HILBERT, D. Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem). Math. Ann., 67, 1909, 281–300. Reprinted in Gesammelte Abhandlungen, Vol. I, 510–527, 2nd edition, Chelsea, Bronx, N.Y., 1965.

    MathSciNet  MATH  Google Scholar 

  11. LANDAU, E. Handbuch der Lehre von der Verteilung der Primzahlen. Teubner, Leipzig, 1909. Reprinted by Chelsea, Bronx, N.Y., 1974.

    Google Scholar 

  12. LEHMER, D.N. List of Prime Numbers from 1 to 10,006,721. Reprinted by Hafner, New York, 1956.

    Google Scholar 

  13. LEHMER, D.N. Factor Table for the First Ten Millions Containing the Smallest Factor of Every Number not Divisible by 2, 3, 5, or 7, Between the Limits 0 and 10,017,000. Reprinted by Hafner, New York, 1956.

    Google Scholar 

  14. WIEFERICH, A. Beweis des Satzes, dass sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen lässt. Math. Ann., 66, 1909, 95–101.

    MathSciNet  MATH  Google Scholar 

  15. SCHUR, I. Uber die Existenz unendlich vieler Primzahlen in einiger speziellen arithmetischen Progressionen. Sitzungsber. Berliner Math. Ges. 11, 1912, 40–50.

    Google Scholar 

  16. STRIDSBERG, E. Sur la démonstration de M. Hilbert du théorème de Waring. Math. Ann., 72, 1912, 145–152.

    MathSciNet  MATH  Google Scholar 

  17. Coblyn Sur les couples de nombres premiers. Soc. Math. France, C.R., 1913, 55–56.

    Google Scholar 

  18. LITTLEWOOD, J.E. Sur la distribution des nombres premiers. C.R. Acad. Sci. Paris, 158, 1914, 1869–1872.

    MATH  Google Scholar 

  19. WEYL, H. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77, 1916, 313–352.

    MathSciNet  MATH  Google Scholar 

  20. BRUN, V. Le crible d’Eratosthène et le théorème de Goldbach. C.R. Acad. Sci. Paris, 168, 1919, 544–546.

    MATH  Google Scholar 

  21. BRUN, V. La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 +... où les dénominateurs sont “nombres premiers jumeaux” est convergente ou finie. Bull. Sci. Math., (2), 43, 1919, 100–104 and 124–128.

    Google Scholar 

  22. RAMANUJAN, S. A proof of Bertrand’s postulate. J. Indian Math. Soc., 11, 1919, 181–182. Reprinted in Collected Papers (edited by G.H. Hardy, P.V. Seshu Aiyar, & B.M. Wilson), 208–209. Cambridge Univ. Press, Cambridge, 1927. Reprinted by Chelsea, Bronx, N.Y., 1962.

    Google Scholar 

  23. BRUN, V. Le crible d’Erathostène et la théorème de Goldbach. Videnskapsselskapets Skrifter Kristiania, Mat.-nat. Kl. 1920, No. 3, 36 pages.

    Google Scholar 

  24. HARDY, G.H. & LITTLEWOOD, J.E. A new solution of Waring’s problem. Quart. J. Math., 1920, 48, 272–293. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 382–393. Clarendon Press, Oxford, 1966.

    Google Scholar 

  25. HARDY, G.H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, I: A new solution of Waring’s Problem. Nachr. Königl. Ges. d. Wiss. zu Göttingen, 1920, 33–54. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 405–426. Clarendon Press, Oxford, 1966.

    Google Scholar 

  26. HARDY, G.H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, II: Proof that every number is the sum of at most 21 biquadrates. Math. Zeits., 9, 1921, 14–27. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 427–440. Clarendon Press, Oxford, 1966.

    MathSciNet  MATH  Google Scholar 

  27. KAMKE, E. Verallgemeinerung des WaringHilbertschen Satzes. Math. Ann., 83, 1921, 85–112.

    MathSciNet  Google Scholar 

  28. HARDY, G.H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, IV: The singular series in Waring’s Problem and the value of the number G(k). Math. Zeits., 12, 1922, 161–188. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 441–468. Clarendon Press, Oxford, 1966.

    MathSciNet  MATH  Google Scholar 

  29. HARDY, G.H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1–70. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 441–468. Clarendon Press, Oxford, 1966.

    MathSciNet  MATH  Google Scholar 

  30. HARDY, G. H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1–70. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 561–630. Clarendon Press, Oxford, 1966.

    MathSciNet  MATH  Google Scholar 

  31. HARDY, G.H. & Littlewood, J.E. Some problems of “Partitio Numerorum”, III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1–70. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 441–468. Clarendon Press, Oxford, 1966.

    MathSciNet  MATH  Google Scholar 

  32. HARDY, G. H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1–70. Reprinted in Collected Papers of G.H. Hardy, Vol. I, 561–630. Clarendon Press, Oxford, 1966.

    MathSciNet  MATH  Google Scholar 

  33. NAGELL, T. Zahlentheoretische Notizen I, Ein Beitrag zur Theorie der höheren Kongruenzen. Vidensk. Skrifter, ser. I, Math. Nat. Kl., No. 13, Kristiania, 1923, 3–6.

    Google Scholar 

  34. RADEMACHER, H. Beiträge zur Viggo Brunschen Methode in der Zahlentheorie. Abh. Math. Sem. Univ. Hamburg, 3, 1924, 12–30. Reprinted in Collected Papers (edited by E. Grosswald), Vol. I, 280–288. M.I.T. Press, Cambridge, 1974.

    Google Scholar 

  35. SCHOENBERG, I.J. Uber die asymptotische Verteilung reeller Zahlen (mod 1). Math. Zeits., 28, 1928, 171–199.

    MathSciNet  MATH  Google Scholar 

  36. WINTNER, A. Uber den Konvergenzbegriff der mathematischen Statistik. Math. Zeits., 28, 1928, 476–480.

    MathSciNet  MATH  Google Scholar 

  37. ERDöS, P. Beweis eines Satzes von Tschebychef. Acta Sci. Math. Szeged, 5, 1930, 194–198.

    Google Scholar 

  38. HOHEISEL, G. Primzahlprobleme in der Analysis. Sitzungsberichte Berliner Akad. d. Wiss., 1930, 580–588.

    Google Scholar 

  39. SCHNIRELMANN, L. Über additive Eigenschaften von Zahlen. Ann. Inst. Polytechn. Novocerkask, 14, 1930, 3–28 and Math. Ann., 107, 1933, 649–690.

    Google Scholar 

  40. TITCHMARSH, E.C. A divisor problem. Rend. Cir. Mat. Palermo, 54, 1930, 414–429.

    MATH  Google Scholar 

  41. WESTZYNTHIUS, E. Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Comm. Phys. Math. Helsingf ors, (5), 25, 1931, 1–37.

    Google Scholar 

  42. LANDAU, E. Uber den Wienerschen neuen Weg zum Primidealsatz. Sitzungsber. Berliner Akad. d. Wiss., 1932, 514–521.

    Google Scholar 

  43. SKEWES, S. On the difference π(x) — li(x). J. London Math. Soc., 8, 1933, 277–283.

    MathSciNet  Google Scholar 

  44. CHOWLA, S. On the least prime in an arithmetic progression. J. Indian Math. Soc., (2), 1, 1934, 1–3.

    Google Scholar 

  45. ISHIKAWA, H. Über die Verteilung der Primzahlen. Sci. Rep. Tokyo Bunrika Daigaku, A, 2, 1934, 27–40.

    Google Scholar 

  46. ERDOS, P. On the density of some sequences of numbers, I. J. London Math. Soc., 10, 1935, 120–125.

    Google Scholar 

  47. ERDOS, P. On the difference of consecutive primes. Quart. J. Pure & Appl. Math., Oxford, 6, 1935, 124–128.

    Google Scholar 

  48. ERDOS, P. On the normal number of prime factors of p —1 and some related problems concerning Euler’s ϕ-function. Quart. J. Pure & Appl. Math., Oxford, 6, 1935, 205–213.

    Google Scholar 

  49. VINOGRADOV, I.M. On Waring’s problem. Annals of Math., 36, 1935, 395–405.

    MATH  Google Scholar 

  50. DICKSON, L.E. Solution of Waring’s problem. Amer. J. Math., 58, 1936, 530–535. Reprinted in The Collected Mathematical Papers (edited by A.A. Albert), Vol. III, 290–295. Chelsea, Bronx, N.Y., 1975.

    MathSciNet  Google Scholar 

  51. HEILBRONN, H. Uber das Waringsche Problem. Acta Arithm., 1, 1936, 212–221.

    MATH  Google Scholar 

  52. CRAMER, H. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithm., 2, 1937, 23–46.

    Google Scholar 

  53. ERDOS, P. On the density of some sequences of numbers, II. J. London Math. Soc., 12, 1937, 7–11.

    Google Scholar 

  54. INGHAM, A.E. On the difference between consecutive primes. Quart. J. Pure & Appl. Math., Oxford, Ser. 2, 8, 1937, 255–266.

    Google Scholar 

  55. LANDAU, E. Über einige neuere Fortschritte der additiven Zahlentheorie. Cambridge Univ. Press, Cambridge, 1937. Reprinted by Stechert-Hafner, New York, 1964.

    Google Scholar 

  56. TURAN, P. Uber die Primzahlen der arithmetischen Progressionen. Acta Sci. Math. Szeged, 8, 1937, 226–235.

    Google Scholar 

  57. VAN DER CORPUT, Sur l’h ypothese de Goldbach pour J.G. presque tous les nombres pairs. Acta Arithm., 2, 1937, 266–290.

    MATH  Google Scholar 

  58. VINOGRADOV, I.M. Representation of an odd number as the sum of three primes (in Russian). Dokl. Akad. Nauk SSSR, 15, 1937, 169–172.

    MATH  Google Scholar 

  59. VINOGRADOV, I.M. Some new problems in the theory of primes (in Russian). Doklady Akad. Nauk SSSR, 16, 1937, 131–132.

    MATH  Google Scholar 

  60. VINOGRADOV, I.M. Some theorems concerning the theory of primes (in Russian). Mat. Sbornik, N.S., 2, (44), 1937, 179–195.

    MATH  Google Scholar 

  61. ERDOS, P. On the density of some sequences of numbers, III. J. London Math. Soc., 13, 1938, 119–127.

    Google Scholar 

  62. ESTERMANN, T. Proof that almost all even positive integers are sums of two primes. Proc. London Math. Soc., 44, 1938, 307–314.

    MathSciNet  Google Scholar 

  63. HARDY, G.H. & Edmund Landau. Heilbronn, H. J. London Math. Soc., 13, 1938, 302–310. Reprinted in Collected Papers of G.H. Hardy, Vol. VII, 762–770. Clarendon Press, Oxford, 1979.

    Google Scholar 

  64. POULET, P. Table des nombres composes vérifiant le théorème de Fermat pour le module 2, jusqu’ à 100.000.000. Sphinx, 8, 1938, 52–52 Corrections: Math. Comp., 25, 1971, 944–945 and 26, 1972, p. 814.

    Google Scholar 

  65. RANKIN, R.A. The difference between consecutive prime numbers. J. London Math. Soc., 13, 1938, 242–247.

    MathSciNet  Google Scholar 

  66. ROSSER, J.B. The nth prime is greater thannlogn. Proc. London Math. Soc. 45, 1938, 21–44.

    MathSciNet  Google Scholar 

  67. TSCHUDAKOFF, N.G. On the density of the set of even integers which are not representable as a sum of two odd primes (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 1, 1938, 25–40.

    Google Scholar 

  68. VINOGRADOV, I.M. Some general theorems about primes (in Russian). Tray. Inst. Math. Tbilissi, 3, 1938, 35–67.

    Google Scholar 

  69. DAVENPORT, H. On Waring’s problem for cubes. Acta Math., 71, 1939, 123–143.

    MathSciNet  Google Scholar 

  70. DAVENPORT, H. On Waring’s problem for fourth powers. Annals of Math., 40, 1939, 731–747.

    MathSciNet  Google Scholar 

  71. DICKSON, L.E. All integers, except 23 and 239, are sums of 8 cubes. Bull. Amer. Math. Soc., 45, 1939, 588–591. Reprinted in The Collected Mathematical Papers (edited by A.A. Albert), Vol. V, 66–69. Chelsea, Bronx, N.Y., 1975.

    MathSciNet  Google Scholar 

  72. VAN DER CORPUT, Über Summen von Primzahlen J.G. und Primzahlquadraten. Math. Ann., 116, 1939, 1–50.

    MathSciNet  Google Scholar 

  73. ERDÖS, P. The difference of consecutive primes. Duke Math. J., 6, 1940, 438–441.

    Google Scholar 

  74. INGHAM, A.E. On the estimation of N(σ,T). Quart. J. Math., Oxford, 11, 1940, 291–292.

    MathSciNet  Google Scholar 

  75. PILLAI, S.S. On Waring’s problem: g(6) = 73. Proc. Indian Acad. Sci., A, 12, 1940, 30–40.

    MathSciNet  Google Scholar 

  76. NARASIMHAMURTI, V On Waring’s problem for 8th, 9th and 10th powers. J. Indian Math. Soc., 5, 1941, p. 122.

    MathSciNet  MATH  Google Scholar 

  77. DAVENPORT, H. On Waring’s problem for fifth and sixth powers. Amer. J. Math., 64, 1942, 199–207.

    MathSciNet  MATH  Google Scholar 

  78. MANN, H.B. A proof of the fundamental theorem of sums of sets of positive integers. Annals of Math., 43, 1942, 523–527.

    MATH  Google Scholar 

  79. RUBUGUNDAY, R. On g(k) in Waring’s problem. J. Indian Math. Soc., N.S., 6, 1942, 192–198.

    MathSciNet  MATH  Google Scholar 

  80. ARTIN, E. & SCHERK, P. On the sums of two sets of integers. Annals of Math., 44, 1943, 138–142. Reprinted in Collected Papers (edited by S. Lang & J.T. Tate), 346–350. Addison-Wesley, Reading, Mass., 1965.

    MathSciNet  MATH  Google Scholar 

  81. LINNIK, YU.V. An elementary solution of a problem of Waring by Schnirelmann’s method. Mat. Sbornik, 12 (54), 1943, 225–230.

    MathSciNet  Google Scholar 

  82. CHOWLA, S. There exists an infinity of 3-combinations of primes in A. P. Proc. Lahore Phil. Soc., 6, 1944, 15–16.

    MathSciNet  Google Scholar 

  83. LINNIK, YU.V. On the least prime in an arithmetic progression I. The basic theorem (in Russian). Mat. Sbornik, 15 (57), 1944, 139–178.

    MathSciNet  Google Scholar 

  84. NIVEN, I. An unsolved case of the Waring problem. Amer. J. Math., 66, 1944, 137–143.

    MathSciNet  MATH  Google Scholar 

  85. ERDOS, R. Some remarks on Euler’s ϕ-function and some related problems. Bull. Amer. Math. Soc., 51, 1945, 540–544.

    MathSciNet  Google Scholar 

  86. BRAUER, A. On the exact number of primes below a given limit. Amer. Math. Monthly, 9, 1946, 521–523.

    MathSciNet  Google Scholar 

  87. JOFFE, S.A. Review of Kulik’s “Magnus Canon Divisorum...” Math. Comp., 2, 1946/1947, 139–140.

    Google Scholar 

  88. KHINCHIN, A. YA. Three Pearls of Number Theory. Original Russian edition in Ogiz, Moscow, 1947. Translation into English published by Graylock Press, Baltimore, 1952.

    Google Scholar 

  89. RÉNYI, A. On the representation of even numbers as the sum of a prime and an almost prime. Dokl. Akad. Nauk SSSR, 56, 1947, 455–458.

    MATH  Google Scholar 

  90. VINOGRADOV, I.M. The method of trigonometrical sums in the theory of numbers (in Russian). Tray. Inst. Math. Steklov, 23, 1947, 109 pages.

    Google Scholar 

  91. ERDÖs, P. & TURáN, P. On some new questions on the distribution of prime numbers. Bull. Amer. Math. Soc., 54, 1948, 371–378.

    MathSciNet  MATH  Google Scholar 

  92. VINOGRADOV, I.M. Uber die Abschätzung trigonometrischer Summen mit Primzahlen. Izv. Akad. Nauk SSSR, Ser. Mat., 12, 1948, 225–248.

    MATH  Google Scholar 

  93. CLEMENT, P.A. Congruences for sets of primes. Amer. Math. Monthly, 56, 1949, 23–25.

    MathSciNet  MATH  Google Scholar 

  94. ERDæS, P. Problems and results on the difference of consecutive primes. Publ. Math. Debrecen, 1, 1949, 33–37.

    MathSciNet  Google Scholar 

  95. ERDÖS, P. On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A., 35, 1949, 374–384.

    MathSciNet  MATH  Google Scholar 

  96. ERDÖS, P. On the converse of Fermat’s theorem. Amer. Math. Monthly, 56, 1949, 623–624.

    MathSciNet  MATH  Google Scholar 

  97. MOSER, L. A theorem on the distribution of primes. Amer. Math. Monthly, 56, 1949, 624–625.

    MathSciNet  MATH  Google Scholar 

  98. RICHERT, H.E. Uber Zerfällungen in ungleiche Primzahlen. Math. Zeits., 52, 1949, 342–343.

    MathSciNet  MATH  Google Scholar 

  99. SELBERG, A. An elementary proof of the prime number theorem. Annals of Math., 50, 1949, 305–313.

    MathSciNet  MATH  Google Scholar 

  100. SELBERG, A. An elementary proof of Dirichlet’s theorem about primes in an arithmetic progression. Annals of Math., 50, 1949, 297–304.

    MathSciNet  MATH  Google Scholar 

  101. SELBERG, A. An elementary proof of the prime number theorem for arithmetic progressions. Can. J. Math., 2, 1950, 66–78.

    MathSciNet  MATH  Google Scholar 

  102. ERDOS, P. On some application of Brun’s method. Acta Sci. Math. Szeged, 13, 1950, 57–63.

    Google Scholar 

  103. ERDÖS, P. On almost primes. Amer. Math. Monthly, 57, 1950, 404–407.

    MathSciNet  MATH  Google Scholar 

  104. HASSE, H. Vorlesungen über Zahlentheorie. Springer-Verlag, Berlin, 1950.

    MATH  Google Scholar 

  105. SELBERG, A. The general sieve method and its place in prime number theory. Proc. Int. Congr. Math., Cambridge,

    Google Scholar 

  106. SHAPIRO, H.N. On a theorem of Selberg and generalization. Annals of Math., (2), 51, 1950, 485–497.

    MATH  Google Scholar 

  107. NAGELL, T. Introduction to Number Theory. Almqvist & Wiksell, Stockholm, 1964. Reprinted by Chelsea, Bronx, N.Y., 1964.

    Google Scholar 

  108. TITCHMARSH, E.C. The Theory of the Riemann ZetaFunction. Clarendon Press, Oxford, 1951.

    Google Scholar 

  109. WATSON, G.L. A proof of the seven cube theorem. J. London Math. Soc., 26, 1951, 153–156.

    MathSciNet  MATH  Google Scholar 

  110. NAGURA, J. On the interval containing at least one prime number. Proc. Japan Acad., 28, 1952, 177–181.

    MathSciNet  MATH  Google Scholar 

  111. PYATETSKII-SHAPIRO, I.I. On the distribution of prime numbers in sequence of the form [f(n)] (in Russian). Mat. Sbornik, N.S. 33 (75), 1953, 559–566.

    MathSciNet  Google Scholar 

  112. RIEGER, G.J. Zur Hilbertschen Lösung der Waringschen Problems: Abschätzung von g(n). Arch. d. Math., 4, 1953, 275–281.

    MathSciNet  MATH  Google Scholar 

  113. BREUSCH, R. Another proof of the prime number theorem. Duke M.J., 21, 1954, 49–53.

    MathSciNet  MATH  Google Scholar 

  114. RIEGER, G.J. Zu Linniks Lösung des Waringsches Problems: Abschätzung von g(n). Math. Zeits., 60, 1954, 213–234.

    MathSciNet  MATH  Google Scholar 

  115. FJELLSTEDT, L. Bemerkungen über gleichzeitige Lösbarkeit von Kongruenzen. Arkiv Mat., 3, 1955, 193–198.

    MathSciNet  MATH  Google Scholar 

  116. RICCI, G. Recherches sur l’allure de la suite {(pn+1-pn)/log pn}. Coll. Th. Nombres Bruxelles 1955, 93–106. G. Thone, Liège, 1956.

    Google Scholar 

  117. SKEWES, S. On the difference π(x) - li(x), II. Proc. London. Math. Soc., 5, 1955, 48–70.

    MathSciNet  MATH  Google Scholar 

  118. AMITSUR, S.A. On arithmetic functions. J. Anal. Math., 5, 1956/7, 273–314.

    MathSciNet  Google Scholar 

  119. ERDOS, P. On pseudo-primes and Carmichael numbers. Publ. Math. Debrecen, 4, 1956, 201–206.

    MathSciNet  Google Scholar 

  120. NIVEN, I. Irrational Numbers. Carus Math. Monographs, No. 11. Math. Assoc. of America, Washington, 1956.

    MATH  Google Scholar 

  121. OSTMANN, H.H. Additive Zahlentheorie (2 volumes). Springer-Verlag, Berlin, 1956 (2nd edition 1969).

    Google Scholar 

  122. HUA, L.K. Additive Theory of Prime Numbers. Inst. Math. Chinese Acad. Sciences, Peking, 1957. Translated into English by H.H. Ng. Amer. Math. Soc., Providence, 1965.

    Google Scholar 

  123. LEECH, J. Note on the distribution of prime numbers. J. London Math. Soc., 32, 1957, 56–58.

    MathSciNet  MATH  Google Scholar 

  124. MAHLER, K. On the fractional parts of powers of real numbers. Matematika, 4, 1957, 122–124.

    MathSciNet  MATH  Google Scholar 

  125. BAKER, C.L. & GRUENBERGER, E.J. Primes in the Thousandth Million. The Rand Corp., Santa Monica, 1958.

    Google Scholar 

  126. CHEN, J.R. On Waring’s problem for nth powers. Acta Math. Sinica, 8, 1958, 253–257;

    MathSciNet  MATH  Google Scholar 

  127. CHEN, J.R. Chinese Math. Acta, 8, 1966/7, 849–853.

    MATH  Google Scholar 

  128. SCHINZEL, A. & SIERPINSKI, W. Sur certaines hypothèses concernant les nombres premiers. Acta Arithm., 4, 1958, 185–208;

    MathSciNet  MATH  Google Scholar 

  129. SCHINZEL, A. & SIERPINSKI, W. Erratum, 5, 1959, p. 259.

    MathSciNet  Google Scholar 

  130. BAKER, C.L. & GRUENBERGER, F.J. The First Six Million Prime Numbers. Microcard Found., Madison, 1959.

    Google Scholar 

  131. HUA, L.K. Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie. Enzykl. d. Math. Wiss., I2, Heft 13, t.l. Teubner, Leipzig, 1959.

    MATH  Google Scholar 

  132. KAC, M. Statistical Independence in Probability, Analysis and Number Theory. Carus Math. Monographs, No. 12, Math. Assoc. of America, Washington, 1959.

    MATH  Google Scholar 

  133. LEHMER, D.H. On the exact number of primes less than a given limit. Illinois J. Math., 3, 1959, 381–388. Reprinted in Selected Papers (edited by D. McCarthy), Vol. III, 1104–1111. Ch. Babbage Res. Centre, St. Pierre, Manitoba, Canada, 1981.

    MathSciNet  MATH  Google Scholar 

  134. SHANKS, D. Quadratic residues and the distribution of primes. Math. Comp., 13, 1959, 272–284.

    MathSciNet  MATH  Google Scholar 

  135. SCHINZEL, A. Démonstration d’une conséquence de l’hypothèse de Goldbach. Compositio Math., 14, 1959, 74–76.

    MathSciNet  Google Scholar 

  136. SCHINZEL, A. Sur une conséquence de l’hypothèse de Goldbach. Izvestija Mat. Inst., Bulgarian Acad. Sci., 4, 1959, 35–38.

    MathSciNet  Google Scholar 

  137. VINOGRADOV, I.M. On an upper bound for G(n) (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 23, 1959, 637–642.

    MathSciNet  MATH  Google Scholar 

  138. JACOBSTHAL, E. Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, I, II, III. Norske Videnskabsselskab Forhdl., 33, 1960, 117–139.

    MathSciNet  MATH  Google Scholar 

  139. NEWMAN, D.J. A simplified proof of Waring’s conjecture. Michigan Math. J., 7, 1960, 291–295.

    MathSciNet  MATH  Google Scholar 

  140. JURKAT, W.B. Eine Bemerkung zur Vermutung von Mertens. Machr. der Osterr. Math. Ges., Sondernummer Ber. V. Osterr. Math.-Kongress, Vienna, 1961, 11.

    Google Scholar 

  141. PRACHAR, K. Uber die kleinste Primzahl einer arithmetischen Reihe. Journal f. d. reine u. angew. Math., 206, 1961, 3–4.

    MathSciNet  MATH  Google Scholar 

  142. ROTKIEWICZ, A. Démonstration arithmétique d’existence d’une infinité de nombres premiers de la forme nk + 1. L’enseign. Math., (2), 7, 1962, 277–280.

    MathSciNet  MATH  Google Scholar 

  143. WRENCH, J.W. Evaluation of Artin’s constant and the twin-prime constant. Math. Comp., 15, 1961, 396–398.

    MathSciNet  MATH  Google Scholar 

  144. ERDÖs, P. On the integers relatively prime to n and on a number theoretic function considered by Jacobsthal. Math. Scand., 10, 1962, 163–170.

    MathSciNet  MATH  Google Scholar 

  145. ROSSER, J.B. & SCHOENFELD, L. Approximate formulas for some functions of prime numbers. Illinois. J. Math., 6, 1962, 64–94.

    MathSciNet  MATH  Google Scholar 

  146. SCHINZEL, A. Remark on a paper of K. Prachar, “Uber die kleinste Primzahl einer arithmetischen Reihe”. Journal f. d. reine u. angew. Math., 210, 1962, 121–122.

    MathSciNet  MATH  Google Scholar 

  147. SEGAL, S. On n(x+y) ≤ π(x) + π(y). Trans. Amer. Math. Soc., 104, 1962, 523–527.

    MathSciNet  MATH  Google Scholar 

  148. AYOUB, R.G. An Introduction to the Theory of Numbers. Amer. Math. Soc., Providence, R.I., 1963.

    MATH  Google Scholar 

  149. ESTERMANN, T. Note on a paper of A. Rotkiewicz. Acta Arithm., 8, 1963, 465–467.

    MathSciNet  MATH  Google Scholar 

  150. KANOLD, H.J. Elementare Betrachtungen zur Primzahltheorie. Arch. Math., 14, 1963, 147–151.

    MathSciNet  MATH  Google Scholar 

  151. NEUBAUER, G. Eine empirische Untersuchung zur Mertenssche Funktion. Numer. Math., 5, 1963, 1–13.

    MathSciNet  MATH  Google Scholar 

  152. RANKIN, R.A. The difference between consecutive prime numbers, V. Proc. Edinburgh Math. Soc., (2), 13, 1963, 331–332.

    MathSciNet  MATH  Google Scholar 

  153. ROTKIEWICZ, A. Sur les nombres pseudo-premiers de la forme ax + b. C.R. Acad. Sci. Paris, 257, 1963, 2601–2604.

    MathSciNet  MATH  Google Scholar 

  154. WALFISZ, A.Z. Weylsche Exponentialsummen in der neueren Zahlentheorie. Veb Deutscher Verlag d. Wiss., Berlin, 1963.

    MATH  Google Scholar 

  155. CHEN, J.R. Waring’s problem for g(5) = 37. Sci. Sinica, 13, 1964, 1547–1568. Reprinted in Chinese Mathematics, 6, 1965, 105–127.

    MathSciNet  Google Scholar 

  156. KANOLD, H.J. Uber Primzahlen in arithmetischen Folgen. Math. Ann., 156, 1964, 393–395

    MathSciNet  MATH  Google Scholar 

  157. KANOLD, H.J. Math. Ann., 157, 1965, 358–362.

    MathSciNet  MATH  Google Scholar 

  158. KAPFERER, H. Verifizierung des symmetrischen Teils der Fermatschen Vermutung für unendlich viele paarweise teilerfremde Exponenten E. Journal f. d. reine u. angew. Math., 214/5, 1964, 360–372.

    MathSciNet  Google Scholar 

  159. ROHRBACH, H. & WEIS, J. Zum finiten Fall des Bertrandschen Postulats. Journal f. d. reine u. angew. Math., 214/5, 1964, 432–440.

    MathSciNet  Google Scholar 

  160. SHEN, M.K. On checking the Goldbach conjecture. Nordisk Tidskr., 4, 1964, 243–245.

    MATH  Google Scholar 

  161. SIERPIńSKI, W. Elementary Theory of Numbers. Haf ner, New York, 1964.

    MATH  Google Scholar 

  162. STEMMLER, R.M. The ideal Waring theorem for exponents 401–200,000. Math. Comp., 18, 1964, 144–146.

    MathSciNet  MATH  Google Scholar 

  163. BATEMAN, P.T. & LOW, M.E. Prime numbers in arithmetic progression with difference 24. Amer. Math. Monthly, 72, 1965, 139–143.

    MathSciNet  MATH  Google Scholar 

  164. GELFOND, A.O. & LINNIK, Yu.V. Elementary Methods in Analytic Number Theory. Translated by A. Feinstein, revised and edited by L.J. Mordell. Rand McNally, Chicago, 1965.

    Google Scholar 

  165. PAN, C.D. On the least prime in an arithmetic progression. Sci. Record (N.S.), 1, 1957, 311–313.

    MathSciNet  MATH  Google Scholar 

  166. ROTKIEWICZ, A. Les intervalles contenant les nombres pseudo premiers. Rend. Circ. Mat. Palermo (2), 14, 1965, 278–280.

    MathSciNet  MATH  Google Scholar 

  167. STEIN, M.L. & STEIN, P.R. New experimental results on the Goldbach conjecture. Math. Mag., 38, 1965, 72–80.

    MathSciNet  MATH  Google Scholar 

  168. STEIN, M.L. & STEIN, P.R. Experimental results on additive 2-bases. Math. Comp., 19, 1965, 427–434.

    MATH  Google Scholar 

  169. BEILER, A.H. Recreations in the Theory of Numbers (The Queen of Mathematics Entertains). Dover, New York, 1966.

    MATH  Google Scholar 

  170. CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao, 17, 1966, 385–386.

    Google Scholar 

  171. BOMBIERI, E. & DAVENPORT, H. Small differences between prime numbers. Proc. Roy. Soc., A, 293, 1966, 1–18.

    MathSciNet  MATH  Google Scholar 

  172. LEHMAN, R.S. On the difference n(x) — li(x). Acta Arithm., 11, 1966, 397–410.

    MathSciNet  MATH  Google Scholar 

  173. JARDEN, D. Existence of arbitrarily long sequences of consecutive numbers in arithmetic progressions divisible by arbitrarily many different primes. Fibonacci Quart., 5, 1967, p. 287.

    Google Scholar 

  174. JONES, M.F., LAL, M. & BLUNDON, W.J. Statistics on certain large primes. Math. Comp., 21, 1967, 103–107.

    MathSciNet  MATH  Google Scholar 

  175. KOLESNIK, G.A. The distribution of primes in sequences of the form [n c ] (in Russian). Mat. Zametki, 2, 1967, 117–128.

    MathSciNet  MATH  Google Scholar 

  176. LANDER, L.J. & PARKIN, T.R. Consecutive primes in arithmetic progression. Math. Comp., 21, 1967, p. 489.

    MATH  Google Scholar 

  177. ROTKIEWICZ, A. On the pseudo-primes of the form ax + b Proc. Cambridge Phil. Soc., 63, 1967, 389–392.

    MathSciNet  MATH  Google Scholar 

  178. SZYMICZEK, K. On pseudo-primes which are products of distinct primes. Amer. Math. Monthly, 74, 1967, 35–37.

    MathSciNet  MATH  Google Scholar 

  179. HALBERSTAM, H. & Rotkiewicz, A. A gap theorem for pseudoprimes in arithmetic progressions. Acta Arithm., 13, 1968, 395–404.

    MathSciNet  MATH  Google Scholar 

  180. GRÖLZ, W. Primteiler von Polynomen. Math. Ann., 181, 1969, 134–136.

    MathSciNet  MATH  Google Scholar 

  181. MONTGOMERY, H.L. Zeros of L-functions. Invent. Math., 8, 1969, 346–354.

    MathSciNet  MATH  Google Scholar 

  182. NAGELL, T. Sur les diviseurs premiers des polynômes. Acta Arithm., 15, 1969, 235–244.

    MathSciNet  MATH  Google Scholar 

  183. RICHERT, H.E. Selberg’s sieve with weights. Mathematika, 16, 1969, 1–22.

    MathSciNet  Google Scholar 

  184. ROSSER, J.B., YOHE, J.M. & SCHOENFELD, L. Rigorous computation of the zeros of the Riemann zeta-function (with discussion). Inform. Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), Vol. I, 70–76. North-Holland, Amsterdam, 1969.

    Google Scholar 

  185. SCHOENFELD, L. An improved estimate for the summatory function of the Möbius function. Acta Arithm., 15, 1969, 221–233.

    MathSciNet  MATH  Google Scholar 

  186. WÓJCYK, J. A refinement of a theorem of Schur on primes in arithmetic progressions, III. Acta Arithm. 15, 1969, 193–197.

    Google Scholar 

  187. DRESSLER, R.E. A density which counts multiplicity. Pacific J. Math., 34, 1970, 371–378.

    MathSciNet  MATH  Google Scholar 

  188. HORNFECK, B. Primteiler von Polynomen. Journal f. d. reine u. angew. Math., 243, 1970, p. 120.

    MathSciNet  MATH  Google Scholar 

  189. MOTOHASHI, Y. A note on the least prime in an arithmetic progression with a prime difference. Acta Arithm., 17, 1970, 283–285.

    MathSciNet  MATH  Google Scholar 

  190. SERRE, J.P. Cours d’Arithmétique. Presses Univ. France, Paris, 1970. English translation published by Springer-Verlag, New York, 1973.

    Google Scholar 

  191. ELLIOTT, P.D.T.A. & HALBERSTAM, H. The least prime in arithmetic progression. Studies in Pure Mathematics (edited by R. Rada), 59–61. Academic Press, London, 1971.

    Google Scholar 

  192. ELLISON, W.J. Waring’s problem. Amer. Math. Monthly, 78, 1971, 10–36.

    MathSciNet  MATH  Google Scholar 

  193. GERST, I. & Brillhart, J. On prime divisors of polynomials. Amer. Math. Monthly, 78, 1971, 250–266.

    MathSciNet  MATH  Google Scholar 

  194. MONTGOMERY, H.L. Topics in Multiplicative Number Theory. Lecture Notes in Math., #227. Springer-Verlag, New York, 1971.

    Google Scholar 

  195. SERGUSOV, I.S.A. On the problem of prime-twins (in Russian). Jaroslay. Gos. Ped. Inst. Ucen. Zap., 82, 1971, 85–86.

    MathSciNet  Google Scholar 

  196. TITCHMARSH, E.C. The Theory of the Riemann Zeta Function. Clarendon Press, Oxford, 1951.

    MATH  Google Scholar 

  197. TURAN, P. On some recent results in the analytical theory of numbers. Proc. Symp. Pure Mathematics (1969 Number Theory Institute), vol. 20, 359–374. Amer. Math. Soc., Providence, R.I., 1971.

    MathSciNet  Google Scholar 

  198. BATEMAN, P.T. The distribution of values of Euler function. Acta Arithm., 21, 1972, 329–345.

    MathSciNet  MATH  Google Scholar 

  199. DESHOUILLERS, J.M. Nombres premiers de la forme [n c ]. C.R. Acad. Sci. Paris, Ser. A, 282, 1976, 131–133.

    MathSciNet  MATH  Google Scholar 

  200. HUXLEY, M.N. On the difference between consecutive primes. Invent. Math., 15, 1972, 164–170.

    MathSciNet  MATH  Google Scholar 

  201. HUXLEY, M.N. The Distribution of Prime Numbers. Oxford Univ. Press, Oxford, 1972.

    MATH  Google Scholar 

  202. ROTKIEWICZ, A. On a problem of W. Sierpiński. Elem. d. Math., 27, 1972, 83–85.

    MathSciNet  MATH  Google Scholar 

  203. WALL, C.R. Density bounds for Euler’s function. Math. Comp., 26, 1972, 779–783.

    MathSciNet  MATH  Google Scholar 

  204. APOSTOL, T.M. Another elementary proof of Euler’s formula for ζ(2n). Amer. Math. Monthly, 80, 1973, 425–431.

    MathSciNet  MATH  Google Scholar 

  205. BRENT, R.P. The first occurrence of certain large prime gaps. Math. Comp., 35, 1980, 1435–1436.

    MathSciNet  MATH  Google Scholar 

  206. CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II. Sci. Sinica, 16, 1973, 157–176;

    MathSciNet  MATH  Google Scholar 

  207. CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II. Sci. Sinica, 21, 1978, 421–430.

    MathSciNet  MATH  Google Scholar 

  208. HENSLEY, D. & RICHARDS, I. Primes in intervals. Acta Arithm., 25, 1973/4, 375–391.

    MathSciNet  Google Scholar 

  209. MONTGOMERY, H.L. The pair correlation of zeros of the zeta function. Analytic Number Theory (Proc. Symp. Pure Math., Vol. XXIV, St. Louis, 1972), 181–193. Amer. Math. Soc., Providence, R.I., 1973.

    Google Scholar 

  210. WUNDERLICH, M.C. On the Gaussian primes on the line Im(x) = 1. Math. Comp., 27, 1973, 399–400.

    MathSciNet  MATH  Google Scholar 

  211. AYOUB, R.G. Euler and the zeta function. Amer. Math. Monthly, 81, 1974, 1067–1086.

    MathSciNet  MATH  Google Scholar 

  212. BRENT, R.P. The distribution of small gaps between successive primes. Math. Comp., 28, 1974, 315–324.

    MathSciNet  MATH  Google Scholar 

  213. EDWARDS, H.M. Riemann’s Zeta Function. Academic Press, New York, 1974.

    MATH  Google Scholar 

  214. HALBERSTAM, H. & RICHERT, H.E. Sieve Methods. Academic Press, New York, 1974.

    MATH  Google Scholar 

  215. LEVINSON, N. More than one third of zeros of Riemann’s zeta function are on σ = 1/2. Adv. in Math., 13, 1984, 383–436.

    MathSciNet  Google Scholar 

  216. MAKOWSKI, A. On a problem of Rotkiewicz on pseudo-primes. Elem. d. Math., 29, 1974, p. 13.

    Google Scholar 

  217. SHANKS, D. & WRENCH, J.W. Brun’s constant. Math. Comp., 28, 1974, 293–299.

    MathSciNet  MATH  Google Scholar 

  218. BRENT, R.P. Irregularities in the distribution of primes and twin primes. Math. Comp., 29, 1975, 43–56.

    MathSciNet  MATH  Google Scholar 

  219. MONTGOMERY, H.L. & VAUGHAN, R.C. The exceptional set in Goldbach’s problem. Acta Arithm., 27, 1975, 353–370.

    MathSciNet  MATH  Google Scholar 

  220. RAM MURTY, P.M. On the Existence of “Euclidean Proofs” of Dirichlet’s Theorem on Primes in Arithmetic Progressions. B.Sc. Thesis, Carleton University, Ottawa, 1975, 39 pages.

    Google Scholar 

  221. ROSS, P.M. On Chen’s theorem that each large even number has the form p l + p 2 or p 1 + p 2 p 3. J. London Math. Soc., (2), 10, 1975, 500–506.

    MathSciNet  MATH  Google Scholar 

  222. ROSSER, J.B. & Schoenfeld, L. Sharper bounds for Chebyshev functions 8(x) and ψx). Math. Comp., 29, 1975, 243–269.

    MathSciNet  MATH  Google Scholar 

  223. SWIFT, J.D. Table of Carmichael numbers to 109. Math. Comp., 29, 1975, 338–339.

    Google Scholar 

  224. APOSTOL, T.M. Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976.

    MATH  Google Scholar 

  225. BRENT, R.P. Tables concerning irregularities in the distribution of primes and twin primes to 1011. Math. Comp., 30, 1976, p. 379.

    MathSciNet  Google Scholar 

  226. GERIG, S. A simple proof of the prime number theorem. J. Nb. Th., 8, 1976, 131–136.

    MathSciNet  MATH  Google Scholar 

  227. NIVEN, I. & POWELL, B. Primes in certain arithmetic progressions. Amer. Math. Monthly, 83, 1976, 467–469.

    MathSciNet  MATH  Google Scholar 

  228. SCHOENFELD, L. Sharper bounds for Chebyshev functions θ(x) and ψx), II. Math. Comp., 30, 1976, 337–360.

    MathSciNet  MATH  Google Scholar 

  229. VAUGHAN, R.C. A note on Schnirelmann’s approach to Goldbach’s problem. Bull London Math. Soc., 8, 1976, 245–250.

    MathSciNet  MATH  Google Scholar 

  230. DESHOUILLERS, J.M. Sur la constante de Schnirelmann. Sém. Delange-Pisot-Poitou, 17e année, 1975/6, fasc. 2, exp. No. G16, 6 p., Paris, 1977.

    MathSciNet  Google Scholar 

  231. HUDSON, R.H. A formula for the exact number of primes below a given bound in any arithmetic progression. Bull. Austral. Math. Soc., 16, 1977, 67–73.

    MathSciNet  MATH  Google Scholar 

  232. HUDSON, R.H. & BRAUER, A. On the exact number of primes in the arithmetic progressions 4n ± 1 and 6n ± 1. Journal f. d. reine u. angew. Math., 291, 1977, 23–29.

    MathSciNet  MATH  Google Scholar 

  233. HUXLEY, M.N. Small differences between consecutive primes, II. Mathematika, 24, 1977, 142–152.

    MathSciNet  Google Scholar 

  234. JUTILA, M. On Linnik’s constant. Math. Scand., 41, 1977, 45–62.

    MathSciNet  MATH  Google Scholar 

  235. JUTILA, M. Zero-density estimates for L-functions. Acta Arithm., 32, 1977, 52–62.

    Google Scholar 

  236. KUMAR MURTY, V. The Least Prime in an Arithmetical Progression and an Estimate of Linnik’s Constant. B.Sc. Thesis, Carleton Univ., Ottawa, 1977, 45 pages.

    Google Scholar 

  237. LANGEVIN, M. Methodes elementaires en vue du théorèeme de Sylvester. Sém. Delange-Pisot-Poitou, 17e année, 1975/76, fasc. 1, exp. No. G12, 9 pages, Paris, 1977.

    Google Scholar 

  238. POWELL, B. Proof of a special case of Dirichlet’s theorem. Fibonacci Quart., 15, 1977, 167–169.

    Google Scholar 

  239. SMALL, C. Waring’s problem. Math. Mag., 50, 1977, 12–16.

    MathSciNet  MATH  Google Scholar 

  240. WEINTRAUB, S. Seventeen primes in arithmetic progression. Math. Comp., 31, 1977, p. 1030.

    MathSciNet  MATH  Google Scholar 

  241. ZAGIER, D. The first 50 million prime numbers. Math. Intelligencer, Vol. 1, 1977, 7–19. Reprinted in German in Lebendige Zahlen, by W. Borho, J.C. Jantzen, H. Kraft, J. Rohlfs, D. Zagier. Birkhäuser, Basel, 1981.

    Google Scholar 

  242. BAYS, C. & HUDSON, R.H. Details of the first region of integers x with π 3,2 (x) < π 3,1 (x). Math. Comp., 32, 1978, 571–576.

    MathSciNet  MATH  Google Scholar 

  243. BAYS, C. & HUDSON, R.H. On the fluctuations of Littlewood for primes of the form 4n ± 1. Math. Comp., 32, 1941, 281–286.

    MathSciNet  Google Scholar 

  244. ELLISON, W.J. & x, F. Théorie des Nombres. Abrégé d’Histoire des Mathématiques, Vol. I, Chapter V, §vi (edited by J. Dieudonné). Hermann, Paris, 1978.

    Google Scholar 

  245. HEATH-BROWN, D.R. Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Phil. Soc., 83, 1978, 357–375.

    MathSciNet  MATH  Google Scholar 

  246. IWANIEC, H. On the problem of Jacobsthal. Demo. Math., 11, 1978, 225–231.

    MathSciNet  MATH  Google Scholar 

  247. WAGSTAFF, S.S.Jr., The least prime in arithmetic progression with prime difference. Journal f. d. reine u. angew. Math., 301, 1978, 114–115.

    MathSciNet  MATH  Google Scholar 

  248. ATKIN, A.O.L. & RICKERT, N.W. On a larger pair of twin primes. Abstract 79T-A132, Notices Amer. Math. Soc., 26, 1979, A-373.

    Google Scholar 

  249. BALASUBRAMANIAN, R. On Waring’s problem: g(4)≤ 21. Hardy & Ramanujan J., 2, 1979, 31 pages.

    MathSciNet  Google Scholar 

  250. BAYS, C. & HUDSON, R.H. Numerical and graphical description of all axis crossing regions for the moduli 4 and 8 which occur before 1012. Intern. J. Math. & Math. Sci., 2, 1979, 111–119.

    MathSciNet  MATH  Google Scholar 

  251. CHEN, J.R. On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s L-functions, II. Sci. Sinica, 22, 1979, 859–889.

    MathSciNet  MATH  Google Scholar 

  252. ELLIOTT, P.D.T.A. Probabilistic Number Theory (in 2 volumes). Springer-Verlag, New York, 1979.

    MATH  Google Scholar 

  253. GROSSWALD, E. & HAGIS, P.Jr., Arithmetic progressions consisting only of primes. Math. Comp., 33, 1979, 1343–1352.

    MathSciNet  MATH  Google Scholar 

  254. HEATH-BROWN, D.R. & Iwaniec, H. On the difference between consecutive powers. Bull. Amer. Math. Soc., N.S., 1, 1979, 758–760.

    MathSciNet  MATH  Google Scholar 

  255. HLAWKA, E. Theorie der Gleichverteilung. Bibliographisches Institut, Zürich, 1979.

    MATH  Google Scholar 

  256. IWANIEC, H. & JUTILA, M. Primes in short intervals. Arkiv f. Mat., 17, 1979, 167–176.

    MathSciNet  MATH  Google Scholar 

  257. POMERANCE, C. The prime number graph. Math. Comp., 33, 1979, 399–408;.

    MathSciNet  MATH  Google Scholar 

  258. ROTKIEWICZ, A. & WASEN, R. On a number-theoretical series. Publ. Math. Debrecen, 26, 1979, 1–4.

    MathSciNet  MATH  Google Scholar 

  259. WAGSTAFF, S.S.Jr., Greatest of the least primes in arithmetic progressions having a given modulus. Math. Comp., 33, 1979, 1073–1080.

    MathSciNet  MATH  Google Scholar 

  260. WOOLDRIDGE, K. Values taken many times by Euler’s phi-function. Proc. Amer. Math. Soc., 76, 1979, 229–234.

    MathSciNet  MATH  Google Scholar 

  261. BRENT, R.P. The first occurrence of certain large prime gaps. Math. Comp., 35, 1980, 1435–1436.

    MathSciNet  MATH  Google Scholar 

  262. CHEN, J.R. & PAN, C.D. The exceptional set of Goldbach numbers, I. Sci. Sinica, 23, 1980, 416–430.

    MathSciNet  Google Scholar 

  263. ERDös, P. & STRAUS, E.G. Remarks on the difference between consecutive primes. Elem. d. Math., 35, 1980, 115–118.

    MATH  Google Scholar 

  264. IVIĆ, A. Exponent pairs and the zeta-function of Riemann. Studia Sci. Math. Hung., 15, 1980, 157–181.

    MATH  Google Scholar 

  265. KUTSUNA, M. On a criterion for the class number of a quadratic number field to be one. Nagoya Math. J., 79, 1980, 123–129.

    MathSciNet  MATH  Google Scholar 

  266. LIGHT, W.A., FORREST, J., HAMMOND, N., & ROE, S. A note on Goldbach’s conjecture. BIT, 20, 1980, p. 525.

    MathSciNet  MATH  Google Scholar 

  267. NEWMAN, D.J. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87, 1980, 693–696.

    MathSciNet  MATH  Google Scholar 

  268. PINTZ, J. On Legendre’s prime number formula. Amer. Math. Monthly, 87, 1980, 733–735.

    MathSciNet  MATH  Google Scholar 

  269. POMERANCE, C. Popular values of Euler’s function. Mathematika, 27, 1980, 84–89.

    MathSciNet  MATH  Google Scholar 

  270. POMERANCE, C. A note on the least prime in an arithmetic progression. J. Nb. Th., 12, 1980, 218–223.

    MathSciNet  MATH  Google Scholar 

  271. POMERANCE, C., SELFRIDGE, J.L., & x, S.S.Jr., The pseudo primes to 25 • 109. Math. Comp., 35, 1980, 1003–1026.

    MathSciNet  MATH  Google Scholar 

  272. POWELL, B. Problem E2844 (Difference between consecutive primes). Amer. Math. Monthly, 87, 1980, p. 577; 90, 1983, p. 286.

    Google Scholar 

  273. Vanden EYNDEN, C. Proofs that Σ 1/p diverges. C. Amer. Math. Monthly, 87, 1980, 394–397.

    MathSciNet  MATH  Google Scholar 

  274. Van Der POORTEN, A.J. & ROTKIEWICZ, A. On strong pseudoprimes in arithmetic progressions. J. Austral. Math. Soc., A, 29, 1980, 316–321.

    MATH  Google Scholar 

  275. WAGSTAFF, S.S. Jr., Greatest of the least primes in arithmetic progressions having a given modulus. Math. Comp., 33, 1979, 1073–1080.

    MathSciNet  MATH  Google Scholar 

  276. BOHMAN, J. & FRÆBERG, C.E. Numerical investigations of Waring’s problem for cubes. BIT 21, 1981, 118–122.

    MathSciNet  MATH  Google Scholar 

  277. GRAHAM, S. On Linnik’s constant. Acta Arithm., 39, 1981, 163–179.

    MATH  Google Scholar 

  278. HEATH-BROWN, D.R. Three primes and an almost prime in arithmetic progression. J. London Math. Soc., (2), 23, 1981, 396–414.

    MathSciNet  MATH  Google Scholar 

  279. LEAVITT, W.G. & MULLIN, A.A. Primes differing by a fixed integer. Math. Comp., 37, 1981, 581–585;.

    MathSciNet  MATH  Google Scholar 

  280. LEHMER, D.H. On Fermat’s quotient, base two. Math. Comp., 36, 1981, 289–290.

    MathSciNet  MATH  Google Scholar 

  281. MAIER, H. Chains of large gaps between consecutive primes. Adv. in Math., 39, 1981, 257–269.

    MathSciNet  MATH  Google Scholar 

  282. PINTZ, J. On primes in short intervals, I. Studia Sci. Math. Hung., 16, 1981, 395–414.

    MathSciNet  MATH  Google Scholar 

  283. POMERANCE, C. On the distribution of pseudo-primes. Math. Comp., 37, 1981, 587–593.

    MathSciNet  MATH  Google Scholar 

  284. WEINTRAUB, S. A large prime gap. Math. Comp., 36, 1981, p. 279.

    MathSciNet  MATH  Google Scholar 

  285. DIAMOND, H.G. Elementary methods in the study of the distribution of prime numbers. Bull. Amer. Math. Soc., 7, 1982, 553–589.

    MathSciNet  MATH  Google Scholar 

  286. NAIR, M. On Chebyshev type inequalities for primes. Amer. Math. Monthly, 81, 1982, 126–129.

    Google Scholar 

  287. POMERANCE, C. A new lower bound for the pseudoprimes counting function. Illinois J. Math., 26, 1982, 4–9.

    MathSciNet  MATH  Google Scholar 

  288. PRITCHARD, P.A. 18 primes in arithmetic progression. J. Recr. Math., 15, 1982/3, p. 288.

    Google Scholar 

  289. ROMANI, F. Computations concerning Waring’s problem for cubes. Calcolo, 19, 1982, 415–431.

    MathSciNet  MATH  Google Scholar 

  290. THANIGASALAM, K. Some new estimates for G(k) in Waring’s problem. Acta Arithm., 42, 1982/3, 73–78.

    MathSciNet  MATH  Google Scholar 

  291. WILLIAMS, H.C. A note on the Fibonacci quotient Fp-ε/p Can. Math. Bull., 25, 1982, 366–370.

    MATH  Google Scholar 

  292. CHEN, J.R. The exceptional value of Goldbach numbers, II. Sci. Sinica, Ser. A, 26, 1983, 714–731.

    MathSciNet  MATH  Google Scholar 

  293. CONREY, J.B. Zeros of derivatives of Riemann’s xi-function on the critical line. J. Nb. Th., 16, 1983, 49–74.

    MathSciNet  MATH  Google Scholar 

  294. FOUVRY, E. & IWANIEC, H. Primes in arithmetic progressions. Acta Arithm., 42, 1983, 197–218.

    MathSciNet  MATH  Google Scholar 

  295. IVIć, A. Topics in Recent Zeta Function Theory. Publ. Math. d’Orsay, Univ. Paris-Sud. 1983.

    Google Scholar 

  296. KELLER, W. Large twin prime pairs related to Mersenne numbers. Abstracts Amer. Math. Soc., 4, 1983, p. 490.

    Google Scholar 

  297. NICOLAS, J.L. Petites valeurs de la fonction d’Euler. J. Nb. Th., 17, 1983, 375–388.

    MathSciNet  MATH  Google Scholar 

  298. NICOLAS, J.L. Distribution de valeurs de la fonction d’Euler. In Algorithmique, Calcul Formel Arithmétique, exposé 24, 7 pages. Univ. Saint-Etienne, 1983. Reprinted in L’enseign. Math. 30, 1984, 331–338.

    Google Scholar 

  299. POWELL, B. Problem 6429 (Difference between consecutive primes). Amer. Math. Monthly, 90, 1983, p. 338.

    Google Scholar 

  300. RIESEL, H. & VAUGHAN, R.C. On sums of primes. Arkiv f. Mat., 21, 1983, 45–74.

    MathSciNet  MATH  Google Scholar 

  301. ROBIN, G. Estimation de la fonction de Tschebychef θ sur le k-ième nombre premier et grandes valeurs de la fonction ω(n), nombre de diviseurs premiers de n. Acta Arithm., 42, 1983, 367–389.

    MathSciNet  MATH  Google Scholar 

  302. BALASUBRAMANIAN, R. & MOZZOCHI, C.J. An improved upper bound for G(k) in Waring’s problem for relatively small k. Acta Arithm., 63, 1984, 283–285.

    MathSciNet  Google Scholar 

  303. DABOUSSI, H. Sur le theoreme des nombres premiers. C.R. Acad. Sci. Paris, Ser. I, 298, 1984, 161–164.

    MathSciNet  Google Scholar 

  304. DAVIES, R.O. Solution of problem 6429. Amer. Math. Monthly, 91, 1984, p. 64.

    MathSciNet  Google Scholar 

  305. GUPTA, R. & RAM MURTY, P.M. A remark on Artin’s conjecture. Invent. Math., 78, 1984, 127–130.

    MathSciNet  MATH  Google Scholar 

  306. IWANIEC, H. & PINTZ, J. Primes in short intervals. Monatsh. Math., 98, 1984, 115–143.

    MathSciNet  MATH  Google Scholar 

  307. PINTZ, J. On primes in short intervals, II. Stud. Sci. Math. Hung., 19, 1984, 89–96.

    MathSciNet  MATH  Google Scholar 

  308. POWELL, B. & SHAFER, R.E. Solution of problem E 2844. Amer. Math. Monthly, 91, 1984, 310–311.

    MathSciNet  Google Scholar 

  309. SCHROEDER, M.R. Number Theory in Science and Communication. Springer-Verlag, New York, 1984.

    MATH  Google Scholar 

  310. WANG, Y. Goldbach Conjecture. World Scientific Publ., Singapore, 1984.

    MATH  Google Scholar 

  311. BALASUBRAMANIAN, R., CONREY, J.B., & HEATH-BROWN, D.R. Asymptotic mean square of the product of the Riemann zeta- function and a Dirichlet polynomial. Journal f. d. reine u. angew. Math., 357, 1985, 161–181.

    MathSciNet  MATH  Google Scholar 

  312. FOUVRY, E. Theoreme de Brun-Titchmarsh, application au théorème de Fermat. Invent. Math., 79, 1985, 383–407.

    MathSciNet  MATH  Google Scholar 

  313. HUDSON, R.H. Averaging effect on irregularities in the distribution of primes in arithmetic progressions. Math. Comp., 44, 1985, 561–571.

    MathSciNet  MATH  Google Scholar 

  314. IVIĆ, A. The Riemann Zeta-Function. J. Wiley &; Sons, New York, 1985.

    Google Scholar 

  315. LAGARIAS, J.C., MILLER, V.S. & u, A.M. Computing π(x) The Meissel-Lehmer method. Math. Comp., 44, 1985, 537–560.

    MathSciNet  MATH  Google Scholar 

  316. MAIER, H. Small differences between prime numbers. Michigan Math. J., 32, 1985, 221–225.

    MathSciNet  MATH  Google Scholar 

  317. ODLYZKO, A.M. & Te RIELE, H.J.J. Disproof of the Mertens conjecture. Journal f. d. reine u. angew. Math., 357, 1985, 138–160.

    MATH  Google Scholar 

  318. PINTZ, J. An effective disproof of the Mertens conjecture. Preprint No. 55/1985, 9 pages. Math. Inst. Hungarian Acad. Sci., Budapest.

    Google Scholar 

  319. POWELL, B. Problem 1207 (A generalized weakened Goldbach theorem). Math. Mag., 58, 1985, p. 46; 59, 1986, 48–49.

    MathSciNet  Google Scholar 

  320. PRITCHARD, P.A. Long arithmetic progressions of primes; some old, some new. Math. Comp., 45, 1985, 263–267.

    MathSciNet  MATH  Google Scholar 

  321. THANIGASALAM, K. Improvement on Davenport’s iterative method and new results in additive number theory, I and II (Proof that G(5) ≤ 22). Acta Arithm., 46, 1985, 1–31 and 91–112.

    MathSciNet  MATH  Google Scholar 

  322. BALASUBRAMANIAN, R., DESHOUILLERS, J.M. & DRESS, F. PROBLème de Waring pour les bicarrés, 2: résultats auxiliaires pour le theóreme asymptotique. C.R. Acad. Sci. Paris, Ser I, 303, 1986, 161–163.

    MathSciNet  MATH  Google Scholar 

  323. BOMBIERI, E., FRIEDLANDER, J.B. & IWANIEC, H. Primes in arithmetic progression to large moduli, I. Acta Math., 156, 1986, 203–251.

    MathSciNet  MATH  Google Scholar 

  324. COSTA PEREIRA, N. Sharp elementary estimates for the sequence of primes. Port. Math., 43, 1986, 399–406.

    MATH  Google Scholar 

  325. FINN, M.V. & FROHLIGER, J.A. Solution of problem 1207. Math. Mag., 59, 1986, 48–49.

    Google Scholar 

  326. MOZZOCHI, C.J. On the difference between consecutive primes. J. Nb. Th., 24, 1986, 181–187.

    MathSciNet  Google Scholar 

  327. PINTZ, J. A note on the exceptional set in Goldbach’s problem. Math. Institute Hungarian Acad. Sci., Preprint No. 14/1986.

    Google Scholar 

  328. TE RIELE, H.J.J. On the sign of the difference π(x) - e(x). Report NM-R8609, Centre for Math. and Comp. Science, Amsterdam, 1986; Math. Comp., 48, 1987, 323–328.

    MathSciNet  MATH  Google Scholar 

  329. VAN DE LUNE, J., TE RIELE, H.J.J., & WINTER, D.T. On the zeros of the Riemann zeta function in the critical strip, IV. Math. Comp., 47, 1986, 667–681.

    Google Scholar 

  330. VAUGHAN, R.C. On Waring’s problem for small exponents. Proc. London Math. Soc., 52, 1986, 445–463.

    MathSciNet  MATH  Google Scholar 

  331. VAUGHAN, R.C. On Waring’s problem for sixth powers. J. London Math. Soc., (2), 33, 1986, 227–236.

    MathSciNet  MATH  Google Scholar 

  332. WAGON, S. Where are the zeros of zeta of s? Math. Intelligencer 8, No. 4, 1986, 57–62.

    MathSciNet  MATH  Google Scholar 

  333. BOMBIERI, E., FRIEDLANDER, J.B. & IWANIEC, H. Primes in arithmetic progressions to large moduli, II. Math. Ann. 277, 1987, 361–393.

    MathSciNet  MATH  Google Scholar 

  334. BOMBIERI, E., Iwaniec, H. On the order of ζ(1/2 + it). Ann. Scuola Norm. Sup. Pisa (to appear).

    Google Scholar 

  335. ODLYZKO, A.M. On the distribution of spacings between zeros of the zeta function. Math. Comp., 48, 1987, 273–308.

    MathSciNet  MATH  Google Scholar 

  336. THANIGASALAM, K. Improvement on Davenport’s iterative method and new results in additive number theory, III. Acta Arithm., 48, 1987, 97–116.

    MathSciNet  MATH  Google Scholar 

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    Paulo Ribenboim

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Ribenboim, P. (1988). How are the Prime Numbers Distributed?. In: The Book of Prime Number Records. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9938-4_5

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