Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Elementary methods in the study of the distribution of prime numbers


Author: Harold G. Diamond
Journal: Bull. Amer. Math. Soc. 7 (1982), 553-589
MSC (1980): Primary 10H15, 10A25
MathSciNet review: 670132
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • S. A. Amitsur, On arithmetic functions, J. Analyse Math. 5 (1956/1957), 273–314. MR 0105396 (21 #4138)
  • S. A. Amitsur, Arithmetic linear transformations and abstract prime number theorems., Canad. J. Math. 13 (1961), 83–109. MR 0124302 (23 #A1616)
  • Emiliano Aparicio Bernardo, Methods for the approximate calculation of the minimum uniform Diophantine deviation from zero on a segment, Rev. Mat. Hisp.-Amer. (4) 38 (1978), no. 6, 259–270 (Spanish). MR 531469 (80i:10049)
  • [Axe] A. Axer, Über einige Grenzwertsätze, Sitzber. Akad. Wiss. Wien, Math.-nat. Kl. 120 (1911), Abt. Ha, 1253-1298.
  • R. G. Ayoub, On Selberg’s lemma for algebraic fields, Canad. J. Math. 7 (1955), 138–143. MR 0065586 (16,450b)
  • Raymond Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067–1086. MR 0360116 (50 #12566)
  • Thøger S. V. Bang, An inequality for real functions of a real variable and its application to the prime number theorem, On Approximation Theory (Proceedings of Conference in Oberwolfach, 1963), Birkhäuser, Basel, 1964, pp. 155–160. MR 0182615 (32 #98)
  • [Ber] J. Bertrand, Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme, J. École Roy. Poly. 18 (1845), 123-140.
  • Harald Bohr, A survey of the different proofs of the main theorems in the theory of almost periodic functions, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1, Amer. Math. Soc., Providence, R. I., 1952, pp. 339–348. MR 0045238 (13,550c)
  • Enrico Bombieri, Sulle formule di A. Selberg generalizzate per classi di funzioni aritmetiche e le applicazioni al problema del resto nel “Primzahlsatz”, Riv. Mat. Univ. Parma (2) 3 (1962), 393–440 (Italian, with English summary). MR 0154860 (27 #4804)
  • E. Bombieri, On the large sieve, Mathematika 12 (1965), 201–225. MR 0197425 (33 #5590)
  • Enrico Bombieri, Le grand crible dans la théorie analytique des nombres, Société Mathématique de France, Paris, 1974 (French). Avec une sommaire en anglais; Astérisque, No. 18. MR 0371840 (51 #8057)
  • Robert Breusch, Another proof of the prime number theorem, Duke Math. J. 21 (1954), 49–53. MR 0068567 (16,904f)
  • Robert Breusch, An elementary proof of the prime number theorem with remainder term., Pacific J. Math. 10 (1960), 487–497. MR 0113854 (22 #4685)
  • [Bru] V. Brun, [1] La série $ \frac 15 + \frac 17 + \frac {1}{11} + \frac {1}{13} + \frac {1}{17} + \frac {1}{19} + \frac {1}{29} + \frac {1}{31} + \frac {1}{41} + \frac {1}{43} + \frac {1}{59} + \frac {1}{61} +\cdots$ où les dénominateurs sont ``nombres premiers jumeaux'' est convergente ou finie, Bull. Sci. Math. (2) 43 (1919), 100-104 and 124-128.
  • [Bru] V. Brun, [2] Le crible d'Eratosthène et le théorème de Goldbach, Vid. Sel. Skr. mat. naturw. I, No. 3 (1920).
  • [Buc] A. A. Buchstab, [1] Asymptotic estimates of a general number theoretic function, Mat. Sb. (N.S.) 2 (44) (1937), 1239-1246. (Russian)
  • A. A. Buhštab, Combinatorial strengthening of the sieve of Eratosthenes method, Uspehi Mat. Nauk 22 (1967), no. 3 (135), 199–226 (Russian). MR 0218326 (36 #1413)
  • D. A. Burgess, On character sums and 𝐿-series, Proc. London Math. Soc. (3) 12 (1962), 193–206. MR 0132733 (24 #A2570)
  • K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York Inc., New York, 1968. MR 0249348 (40 #2593)
  • K. Chandrasekharan, Arithmetical functions, Die Grundlehren der mathematischen Wissenschaften, Band 167, Springer-Verlag, New York-Berlin, 1970. MR 0277490 (43 #3223)
  • [Chb] P. L. Chebyshev, Mémoire sur les nombres premiers, J. de Math. Pures Appl. (1) 17 (1852), 366-390. Also in Mémoires présentés à l'Académie Impériale des sciences de St.-Pétersbourg par divers savants 7 (1854), 15-33. Also in Oeuvres 1 (1899), 49-70.
  • Jing Run Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176. MR 0434997 (55 #7959)
  • [Coh] L. W. Cohen, The annual meeting of the society, Bull. Amer. Math. Soc. 58 (1952), 159-160.
  • J. G. van der Corput, Démonstration élémentaire du théorème sur la distribution des nombres premiers, Scriptum no. 1, Math. Centrum Amsterdam, 1948 (French). MR 0029412 (10,597a)
  • [Crp] J. G. van der Corput, [2] Sur la reste dans la démonstration élémentaire du théorème des nombres premiers, Colloq. sur la Théorie des Nombres (Bruxelles, 1955), Thone, Liège, 1956, 163-182.
  • K. A. Corrádi, A remark on the theory of multiplicative functions, Acta Sci. Math. (Szeged) 28 (1967), 83–92. MR 0211964 (35 #2839)
  • Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931 (82m:10001)
  • [Dia] H. Diamond, [1] Changes of sign of π(x) - li(x), L'Enseignement Mathématique 21 (1975), 1-14.
  • Harold J. Diamond, Chebyshev type estimates in prime number theory, Séminaire de Théorie des Nombres, 1973-1974 (Univ. Bordeaux I, Talence), Exp. No. 24, Centre Nat. Recherche Sci., Talence, 1974, pp. 11 pp. Lab Théorie des Nombres. MR 0392877 (52 #13690)
  • Harold G. Diamond and Paul Erdős, On sharp elementary prime number estimates, Enseign. Math. (2) 26 (1980), no. 3-4, 313–321 (1981). MR 610529 (83i:10055)
  • Harold G. Diamond and Kevin S. McCurley, Constructive elementary estimates for 𝑀(𝑥), Analytic number theory (Philadelphia, PA, 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 239–253. MR 654531 (84b:10061)
  • Harold G. Diamond and John Steinig, An elementary proof of the prime number theorem with a remainder term., Invent. Math. 11 (1970), 199–258. MR 0280449 (43 #6169)
  • [Dic] L. E. Dickson, History of the theory of numbers, Carnegie Inst., Washington, D.C., 1919; reprinted by Chelsea, New York, 1966.
  • [Dir] G. Lejeune Dirichlet, Über die Bestimmung der mittleren Werte in der Zahlentheorie, Abh. Akad. Wiss. Berlin, 1849; 1851, 69-83. Also in Werke, vol. 2, 1897, 49-66.
  • Yoshikazu Eda, On the prime number theorem, Sci. Rep. Kanazawa Univ. 2 (1953), no. 1, 23–33. MR 0074451 (17,587f)
  • H. M. Edwards, Riemann’s zeta function, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 58. MR 0466039 (57 #5922)
  • William John Ellison, Les nombres premiers, Hermann, Paris, 1975 (French). En collaboration avec Michel Mendès France; Publications de l’Institut de Mathématique de l’Université de Nancago, No. IX; Actualités Scientifiques et Industrielles, No. 1366. MR 0417077 (54 #5138)
  • P. Erdös, 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. MR 0029411 (10,595c)
  • P. Erdős, On the distribution of numbers of the form 𝜎(𝑛)/𝑛 and on some related questions, Pacific J. Math. 52 (1974), 59–65. MR 0354601 (50 #7079)
  • [Erd] P. Erdös, [3] Personal communication.
  • Euclid, The thirteen books of Euclid’s Elements translated from the text of Heiberg. Vol. I: Introduction and Books I, II. Vol. II: Books III–IX. Vol. III: Books X–XIII and Appendix, Dover Publications, Inc., New York, 1956. Translated with introduction and commentary by Thomas L. Heath; 2nd ed. MR 0075873 (17,814b)
  • [Eul] L. Euler, Varias observationes circa series infinitas, Comment. Acad. Sci. Imp. Petro-politanae 9 (1737; 1744), 160-188. Also in Opera omnia (1) 14, 216-244.
  • È. K. Fogels, On an elementary proof of the prime number theorem, Latvijas PSR Zinātņu Akad. Fiz. Mat. Inst. Raksti. 2 (1950), 14–45 (Russian, with Latvian summary). MR 0047076 (13,824a)
  • William Forman and Harold N. Shapiro, Abstract prime number theorems, Comm. Pure Appl. Math. 7 (1954), 587–619. MR 0063396 (16,114e)
  • [Gau] C. F. Gauss, Letter to Encke, 24 Dec. 1849, Werke, vol. 2, Kng. Ges. Wiss., Göttingen, 1863, pp. 444-447.
  • A. O. Gel′fond, On the arithmetic equivalent of analyticity of the Dirichlet 𝐿-series on the line 𝑅𝑒𝑠=1, Izv. Akad. Nauk SSSR. Ser. Mat. 20 (1956), 145–166 (Russian). MR 0113850 (22 #4681)
  • [Gel] A. O. Gelfond, [2] Commentary on the papers "On the estimation of the number of primes not exceeding a given value" and "On prime numbers", Collected Works of P. L. Chebyshev, vol. 1, Akad. Nauk SSSR, Moscow-Leningrad, 1946, pp. 285-288. (Russian)
  • A. O. Gel′fond and Yu. V. Linnik, Elementary methods in the analytic theory of numbers, Translated from the Russian by D. E. Brown. Translation edited by I. N. Sneddon. International Series of Monographs in Pure and Applied Mathematics, Vol. 92, Pergamon Press, Oxford-New York-Toronto, Ont., 1966. MR 0201368 (34 #1252)
  • Anthony A. Gioia, The theory of numbers. An introduction, Markham Publishing Co., Chicago, Ill., 1970. MR 0258720 (41 #3366)
  • [Had] J. Hadamard, [1] Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann, J. de Math. Pures Appl. (4) 9 (1893), 171-215; reprinted in Oeuvres de Jacques Hadamard, C.N.R.S., Paris, 1968, vol. 1, pp. 103-147.
  • J. Hadamard, Sur la distribution des zéros de la fonction 𝜁(𝑠) et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896), 199–220 (French). MR 1504264
  • G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403 (German). MR 0230694 (37 #6254)
  • H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR 0424730 (54 #12689)
  • [HaRo] H. Halberstam and K. Roth, Sequences, Oxford, London, 1966.
  • [Har] G. H. Hardy, Prime numbers, Brit. Assn. Rep., 1915, pp. 350-354; also in Collected papers, vol. 2, Oxford Univ. Press, London, 1967, pp. 14-18.
  • G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1–70. MR 1555183, http://dx.doi.org/10.1007/BF02403921
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125 (16,673c)
  • Douglas Hensley and Ian Richards, On the incompatibility of two conjectures concerning primes, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 123–127. MR 0340194 (49 #4950)
  • Douglas Hensley and Ian Richards, Primes in intervals, Acta Arith. 25 (1973/74), 375–391. MR 0396440 (53 #305)
  • A. E. Ingham, The distribution of prime numbers, Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Stechert-Hafner, Inc., New York, 1964. MR 0184920 (32 #2391)
  • [Ing] A. E. Ingham, [2] Note on the distribution of primes, Acta Arith. 1 (1936), 201-211.
  • [Ing] A. E. Ingham, [3] Review of Selberg and Erdös elementary proofs of P.N.T., Math. Reviews 10 (1949), 595-596; Reprinted in Reviews in Number Theory (W. J. LeVeque, ed.), vol. 4, N 20-3, Amer. Math. Soc., Providence, R.I., 1974.
  • Henryk Iwaniec, Rosser’s sieve, Acta Arith. 36 (1980), no. 2, 171–202. MR 581917 (81m:10086)
  • [Jur] W. B. Jurkat, Abstracts of short communications, Proc. Internat. Congr. Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, Sweden, 1963, p. 35.
  • P. Kuhn, Eine Verbesserung des Restgliedes beim elementaren Beweis des Primzahlsatzes, Math. Scand. 3 (1955), 75–89 (German). MR 0074449 (17,587d)
  • [laV] C. J. de la Vallée Poussin, [1] Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Sci. Bruxelles 20 (1896), 183-256.
  • [laV] C. J. de la Vallée Poussin, [2] Sur la fonction ζ (s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée, Memoires Couronnés de l'Acad. Roy des Sciences, Belgique 59 (1899-1900); reprinted in Colloque sur la Théorie des Nombres (Bruxelles, 1955), Thone, Liège, 1956, pp. 9-66.
  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed; With an appendix by Paul T. Bateman. MR 0068565 (16,904d)
  • [Lnd] E. Landau, [2] Über einige neuere Grenzwertsätze, Rend. Cire. Mat. Palermo 34 (1912), 121-131.
  • [Lnd] E. Landau, [3] Vorlesungen über Zahlentheorie, Hirzel, Leipzig, 1927; reprinted by Chelsea, New York, 1947.
  • [Lnd] E. Landau, [4] Über den Wienerschen neuen Weg zum Primzahlsatz, Sitzber. Preuss. Akad. Wiss., 1932, pp. 514-521; reprinted in Handbuch, 1953, pp. 917-924.
  • Michel Langevin, Méthodes élémentaires en vue du théorème de Sylvester, Séminaire Delange-Pisot-Poitou, 17e année: 1975/76. Théorie des nombres: Fasc. 2, Exp. No. G2, Secrétariat Math., Paris, 1977, pp. 9 (French). MR 0450215 (56 #8511)
  • A. F. Lavrik and A. Š. Sobirov, The remainder term in the elementary proof of the prime number theorem, Dokl. Akad. Nauk SSSR 211 (1973), 534–536 (Russian). MR 0323736 (48 #2092)
  • [Leg] A.-M. Legendre, Essai sur la théorie des Nombres, Duprat, Paris, 1798.
  • [Leh] R. S. Lehman, On the difference π(x) - 1i(x), Acta Arith. 11 (1966), 397-410.
  • B. V. Levin and A. S. Faĭnleĭb, Application of certain integral equations to questions of the theory of numbers, Uspehi Mat. Nauk 22 (1967), no. 3 (135), 119–197 (Russian). MR 0229600 (37 #5174)
  • Norman Levinson, A motivated account of an elementary proof of the prime number theorem., Amer. Math. Monthly 76 (1969), 225–245. MR 0241372 (39 #2712)
  • [Lit] J. E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris 158 (1914), 1869-1872.
  • [Man] H. von Mangoldt, [1] Zu Riemanns Abhandlung "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse", J. Reine Angew. Math. 114 (1895), 255-305.
  • H. v. Mangoldt, Zur Verteilung der Nullstellen der Riemannschen Funktion 𝛾(𝑡), Math. Ann. 60 (1905), no. 1, 1–19 (German). MR 1511287, http://dx.doi.org/10.1007/BF01447494
  • [Mat] G. B. Mathews, Theory of numbers. I, Deighton Bell, Cambridge, England, 1892; reprinted by Stechert, New York, 1927.
  • Meissel, Ueber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen, Math. Ann. 2 (1870), no. 4, 636–642 (German). MR 1509683, http://dx.doi.org/10.1007/BF01444045
  • [Mer] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math. 78 (1874), 46-62.
  • Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR 0337847 (49 #2616)
  • Trygve Nagell, Introduction to Number Theory, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. MR 0043111 (13,207b)
  • M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89 (1982), no. 2, 126–129. MR 643279 (83f:10043), http://dx.doi.org/10.2307/2320934
  • Veikko Nevanlinna, Über die elementaren Beweise der Primzahlsätze und deren äquivalente Fassungen, Ann. Acad. Sci. Fenn. Ser. A I No. 343 (1964), 52pp (German). MR 0168539 (29 #5800)
  • [Pin] J. Pintz, Elementary methods in the theory of L-functions. I-VIII, Acta Arith. 31 (1976), 53-60; 31 (1976), 273-289; 31 (1976), 295-306; 31 (1976), 419-429; 32 (1977), 163-171; 32 (1977), 173-178; 32 (1977), 397-406; Corrigendum 33 (1977), 293-295; 33 (1977), 89-98.
  • H. R. Pitt, Tauberian theorems, Tata Institute of Fundamental Research, Monographs on Mathematics and Physics, vol. 2, Oxford University Press, London, 1958. MR 0106376 (21 #5109)
  • G. Pólya, Heuristic reasoning in the theory of numbers, Amer. Math. Monthly 66 (1959), 375–384. MR 0104639 (21 #3392)
  • J. Popken, On convolutions in number theory, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 10–15. MR 0068574 (16,905e)
  • A. G. Postnikov and N. P. Romanov, A simplification of A. Selberg’s elementary proof of the asymptotic law of distribution of prime numbers, Uspehi Mat. Nauk (N.S.) 10 (1955), no. 4(66), 75–87 (Russian). MR 0074450 (17,587e)
  • Karl Prachar, Primzahlverteilung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0087685 (19,393b)
  • [Rie] H.-E. Richert, Lectures on sieve methods, Tata Inst., Bombay, 1976.
  • [Rie] B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Kgl. Preuss. Akad. Wiss. Berlin, 1860, pp. 671-680; also in Werke, 2nd ed., Teubner, Leipzig, 1892, pp. 145-155. Reprinted by Dover, New York, 1953.
  • P. M. Ross, On Chen’s theorem that each large even number has the form 𝑝₁+𝑝₂ or 𝑝₁+𝑝₂𝑝₃, J. London Math. Soc. (2) 10 (1975), no. 4, 500–506. MR 0389816 (52 #10646)
  • 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)
  • 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
  • Lowell Schoenfeld, Sharper bounds for the Chebyshev functions 𝜃(𝑥) and 𝜓(𝑥). II, Math. Comp. 30 (1976), no. 134, 337–360. MR 0457374 (56 #15581b), http://dx.doi.org/10.1090/S0025-5718-1976-0457374-X
  • Wolfgang Schwarz, Einführung in Methoden und Ergebnisse der Primzahltheorie, B.I-Hochschultaschenbücher, vol. 278/278, Bibliographisches Institut, Mannheim-Vienna-Zürich, 1969 (German). MR 0263750 (41 #8350)
  • S. L. Segal, Prime number theorem analogues without primes, J. Reine Angew. Math. 265 (1974), 1–22. MR 0349604 (50 #2097)
  • Atle Selberg, An elementary proof of the prime-number theorem, Ann. of Math. (2) 50 (1949), 305–313. MR 0029410 (10,595b)
  • Atle Selberg, An elementary proof of the prime-number theorem for arithmetic progressions, Canadian J. Math. 2 (1950), 66–78. MR 0033306 (11,419c)
  • Atle Selberg, On elementary methods in primenumber-theory and their limitations, Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949, Johan Grundt Tanums Forlag, Oslo, 1952, pp. 13–22. MR 0053147 (14,726k)
  • Harold N. Shapiro, On a theorem of Selberg and generalizations, Ann. of Math. (2) 51 (1950), 485–497. MR 0033308 (11,419e)
  • Harold N. Shapiro, On primes in arithmetic progression. II, Ann. of Math. (2) 52 (1950), 231–243. MR 0036262 (12,81b)
  • [Sok] A. V. Sokolovski, Letter to H. Diamond and J. Steinig, 17 February, 1973.
  • Wilhelm Specht, Elementare Beweise der Primzahlsätze, Hochschulbücher für Mathematik, Band 30, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956 (German). MR 0086829 (19,250f)
  • J. J. Sylvester, On Tchebycheff’s Theory of the Totality of the Prime Numbers Comprised within Given Limits, Amer. J. Math. 4 (1881), no. 1-4, 230–247. MR 1505291, http://dx.doi.org/10.2307/2369154
  • [Syl] J. J. Sylvester, [2] On arithmetical series, Messenger of Math. (2) 21 (1892), 1-19 and 87-120.
  • Tikao Tatuzawa and Kanesiro Iseki, On Selberg’s elementary proof of the prime-number theorem, Proc. Japan Acad. 27 (1951), 340–342. MR 0046382 (13,725f)
  • E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485 (13,741c)
  • A. I. Vinogradov, The density hypothesis for Dirichet 𝐿-series, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 903–934 (Russian). MR 0197414 (33 #5579)
  • I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Dover Publications, Inc., Mineola, NY, 2004. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport; Reprint of the 1954 translation. MR 2104806 (2005f:11172)
  • Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685 (36 #3737)
  • David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923 (3,232d)
  • [Wie] N. Wiener, [1] A new method in Tauberian theorems, J. Math. Phys. M.I.T. 7 (1927-28), 161-184.
  • Norbert Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. MR 1503035, http://dx.doi.org/10.2307/1968102
  • Norbert Wiener and Leonard Geller, Some prime-number consequences of the Ikehara theorem, Acta Sci. Math. Szeged 12 (1950), no. Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B, 25–28. MR 0034801 (11,644f)
  • [Win] A. Wintner, The theory of measure in arithmetical semigroups, Waverly Press, Baltimore, 1944.
  • Eduard Wirsing, Elementare Beweise des Primzahlsatzes mit Restglied. I, J. Reine Angew. Math. 211 (1962), 205–214 (German). MR 0150116 (27 #119)
  • Eduard Wirsing, Elementare Beweise des Primzahlsatzes mit Restglied. II, J. Reine Angew. Math. 214/215 (1964), 1–18 (German). MR 0166180 (29 #3457)
  • E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen. II, Acta Math. Acad. Sci. Hungar. 18 (1967), 411–467 (German). MR 0223318 (36 #6366)
  • E. M. Wright, The elementary proof of the prime number theorem, Proc. Roy. Soc. Edinburgh. Sect. A. 63 (1952), 257–267. MR 0049218 (14,137d)
  • Don Bernard Zagier, Die ersten 50 Millionen Primzahlen, Birkhäuser Verlag, Basel-Stuttgart, 1977 (German). Elemente der Mathematik Beihefte, No. 15. MR 0480292 (58 #467)

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 10H15, 10A25

Retrieve articles in all journals with MSC (1980): 10H15, 10A25


Additional Information

DOI: http://dx.doi.org/10.1090/S0273-0979-1982-15057-1
PII: S 0273-0979(1982)15057-1