Элементарные методы в изучении распределения простых чисел

G. Diamond

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 45 (1990) no. 2

Bibliographie

[1] Amitsur S. A., “On arithmetic functions”, J. Anal. Math., 5 (1956/57), 273–314 ; “Arithemtic linear transformations and abstract prime number theorems”, Canadian J. Math., 13 (1961), 83–109 ; Canadian J. Math., 21 (1969), 1–5 | DOI | MR | MR | Zbl | MR

[2] Aparicio Bernardo E., “Metodos para et calculo aproximado de la desviacion diofanica, uniforme minima a cero en un segmento”, Rev. Math. Hispanico-Americana, 38 (1978), 259–270 | MR | Zbl

[3] Axer A., “Über einige Grenzwertsätze”, Sitzber. Akad. Wiss. Wien Math.-nat. Kl., 120 (1911), 1253–1298 | Zbl

[4] Ayoub R., “On Selberg's lemma for algebraic fields”, Canad. J. Math., 7 (1955), 137–143 ; “Euler and the zeta function”, Amer. Math. Monthly, 81 (1974), 1067–1086 | MR | DOI | MR | Zbl

[5] Bang Th., “An inequality for real functions of a real variable and its application to the prime number theorem”, Proc. Conf. (Oberwolfach, 1963), Birkhauser, Basel, 1964, 155–160 | MR

[6] Bertrand J., “Memoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme”, J. Ecole Roy. Polyt, 18 (1845), 123–140

[7] Botr H., “Address of Professor Harald Bohr”, Proc. Internat. Congr. Math., Vol. 1 (Cambridge, 1950), Amer. Math. Soc., Providence, RI, 1952, 127–134

[8] Bombien E., “Sulle formule di A. Selberg generalizzate per classi di funzioni aritmetiche e le applicazioni al problema del resto nel “Primzahlsatz””, Riv. Math. Univ. Parma (2), 3 (1962), 393–440 ; “On the large sieve”, Mathematika, 12 (1965), 201–225 ; “Le grand crible dans la théorie analytique des nombres”, Astérisque, 18, Soc. math. de France, Paris, 1974 | MR | MR | MR

[9] Breusch R., “Another proof of the prime number theorem”, Duke Math. J., 21 (1954), 49–53 ; “An elementary proof of the prime number theorem with remainder term”, Pacific. J. Math., 10 (I960), 487–497 | DOI | MR | Zbl | MR | Zbl

[10] Brun V., “La série $1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59=1/61+\dots$ où les dénominateurs sont “nombres premiers jumeaux” est convergente ou finie”, Bull. sci. math. (2), 43 (1919), 100–104, 124–128 ; “Le crible d'Eratostene et le théorème de Goldbach”, Vid. Sel. Skr. mat. naturw., 1:3 (1920) | Zbl

[11] Buxshtab A. A., “Asimptoticheskaya otsenka odnoi obschei teoretiko-chislovoi funktsii”, Mat. sb., 2(44):6 (1937), 1239–1246 ; “Комбинаторное усиление метода эратосфенова решета”, Успехи мат. наук, 22:3 (1967), 199–226 | Zbl | MR

[12] Burgess D. A., “On character sums and $L$ series II”, Proc. London Math. Soc., 12 (1962), 193–206 | DOI | MR | Zbl

[13] Chandrasekharan K., Introduction to analytic number theory, Die Grundlehren der math. Wiss., 148, Springer, New York, 1968 ; Arithmetic functions, Die Grundlehren der math. Wiss., 167, Springer, New York, 1970 ; Chandrasekkharan K., Vvedenie v analiticheskuyu teoriyu chisel, Mir, M., 1974 ; Арифметические функции, Физматгиз, М., 1975 | MR | Zbl | Zbl | MR

[14] Chebyshev P. L., “O prostykh chislakh”, Polnoe sobranie sochinenii P. L. Chebysheva, t. 1, 1944, 191–207

[15] 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”, Sci. Sinica, 16 (1973), 157–176 | MR | Zbl

[16] Cohen L. W., “The annual meeting of the society”, Bull. Amer. Math. Soc., 58 (1952), 159–160 | MR

[17] Corput J. G. van der., “Démonstration élémentaire de théorème sur la distribution des nombres premiers”, Scriptum No. 1, Math. Centrum, Amsterdam, 1948 ; “Sur la reste dans la démonstration élémentaire du théorème des nombres premiers”, Colloq. sur la théorie des nombres (Bruxelles, 1955), Thone, Liege, 1956, 163–182 | Zbl

[18] Corradi K., “A remark on the theory of multiplicative functions”, Acta sci. math. Szeged., 28 (1967), 83–92 | MR | Zbl

[19] Davenport H., Multiplicative number theory, Markham, Chicago, 1967 ; 2nd ed. (revised by H. L. Montgomery), Springer, New York, 1980 ; Devenport G., Multiplikativnaya teoriya chisel, Nauka, M., 1971 | MR | Zbl | MR | Zbl | MR

[20] Diamond H., “Changes of sign of $\pi(x)-\mathrm{li}(x)$”, L'Enseignement math., 21 (1975), 1–14 ; “Chebyshev type estimates in prime number theory”, Exp. Bo. 24, Sem. théorie des nombres, Univ. de Bordeaux I, 1973–74, 11 | MR | Zbl

[21] Diamond H., Erdos P., “On sharp elementary prime number estimates”, L'Enseignement mathematique, 26 (1980), 313–321 | MR | Zbl

[22] Diamond H., McCurley K., “Constructive elementary estimates for $M(x)$”, Proc. Conf. Number Theory (Temple Univ., 1980), Lectures Notes in Math., 899, Springer-Verlag, New York, 1981, 239–253 | MR

[23] Diamond H., Steining H., “An elementary proof of the prime number theorem with a remainder term”, Invent. Math., 11 (1970), 199–258 | DOI | MR

[24] Diskson L. E., History of the theory of numbers, Carnegie Inst., Washington, DC, 1919; Chelsea, New York, 1966

[25] Lejeune Dirichlet G., “Über die Bestimmung der mittleren Werte in der Zahlentheorie”, Abh. Akad. Wiss. Berlin, 1849, 1851, 69–83; also in Werke, 2 (1897), 49–66

[26] Eda Y., “On the prime number theorem”, Sci. Rep. Kanazawa Univ., 2 (1953), 23–33 | MR | Zbl

[27] Edwards H. M., Riemann's zeta function, Academic Press, New York, 1974 | Zbl

[28] Ellison W. J. (in collaboration with M. Mendes – France), Les nombres premiers, Publ. Jnst. Math. Univ. Nancago, IX, Hermann, Paris, 1975 | MR | Zbl

[29] 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 ; “On the distribution of numbers of the form $\sigma(n)/n$ and on some related questions”, Pacific. J. Math., 52 (1974), 59–65 | DOI | MR | Zbl | MR | Zbl

[30] Euclid, Elements, 2, eds. T. Heath, Dover, New York, 1956, IX, 20 ; D. D. Mordukhai-Boltovskaya, Nachala Evklida, T. I, kn. 1–6, Gostekhizdat, M.; L., 1948 | MR

[31] Euler L., “Varias observationes circa series infinitas”, Comment. Acad. Sci. Imp. Petropolitanae, 9 (1737, 1744), 160–188; also in Opera omnia (1), 114, 216–244

[32] Fogels E., “On an elementary proof of the prime number theorem”, Latvijas PSR Zin. Akad. Fiz. Mat. Inst. Raksti, 2 (1950), 14–45 (Russian) | MR | Zbl

[33] Forman W., Shapiro H. N., “Abstract prime number theorems”, Comm. Pure Appl. Math., 7 (1954), 587–619 | DOI | MR | Zbl

[34] Gauss C. F., “Letter to Encke, 24 Dec. 1849”, Werke, 2, Kng Ges. Wiss., Gottingen, 1863, 444–447

[35] Gelfond A. O., “Ob arifmeticheskom ekvivalente analitichnosti $L$-ryada Dirikhle na pryamoi $\mathrm{Re}\,s=1$”, Izv. AN SSSR. Ser. mat., 20 (1956), 145–166 ; “Комментарии к статьям “Об определении числа простых чисел, непревосходящих данной величины” и “О простых числах””, Полное собрание сочинений П. Л. Чебышева, т. 1, АН СССР, М.–Л., 1944, 285–188 | MR | Zbl

[36] Gelfond A. O., Linnik Yu. V., Elementarnye metody v analiticheskoi teorii chisel, Fizmatgiz, M., 1962

[37] Gioja A., The theory of numbers. An introduction, Makkham, Chicago, 1970

[38] Hadamard J., “Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann”, J. math. pures appl. (4), 9 (1893), 171–215; reprinted in Oeuvres de Jacques Hadamard, 1, C.N.R.S., Paris, 1968; “Sur la distribution des zéros de la fonctions $\zeta(s)$ et ses conséquences arithmétiques”, Bull. Soc. Math. France, 24 (1896), 199–220 ; reprinted in Oeuvres, 1, 189–210 | MR | Zbl

[39] Halasz G., “Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen”, Acta math. Acad. sci. Hungar., 19 (1968), 365–403 | DOI | MR | Zbl

[40] Halberstram H., Richert H. E., Sieve methods, Academic Press, London, 1974

[41] Halberstram H., Roth K., Sequences, London, Oxford, 1966

[42] Hardy G. H., “Prime numbers”, Brit. Assn. Rep., 1915, 350–354 ; also in Collected papers, Vol. 2, Univ. Press Oxford, London, 1967, 14–18 | Zbl

[43] 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 | DOI | MR

[44] Hardy G. H., Wright E. M., An introduction to the theory of numers, 5th ed., Oxford, London, 1979 | MR | Zbl

[45] Hensley D., Richards I., “On the incompatibility of two conjectures concerning primes”, Analytic Number Theory, Proc. Sympos. Pure Math., 24, Amer. Math. Soc., Providence, RI, 1973, 123–127 ; “Primes in intervals”, Acta arith., 25 (1974), 375–391 | MR | MR | Zbl

[46] Ingham A. E., The distribution of prime numbers, Cabridge Tracts in Math., 30, Cambridge, 1932 ; reprinted by Hafner, New York, 1971; “Note on the distribution of primes”, Acta arith., 1 (1936), 201–211 ; Ingam A. E., Raspredelenie prostykh chisel, ONTI NKTP SSSR, M.–L., 1936; “Review of Selberg and Erdös elementary proofs of P. N. T.”, Math. Reviews, 10 (1949), 595–596; reprinted in Reviews in Number Theory, 4, no. 20-3, eds. W. J. LeVeque, Amer. Math. Soc., Providence, RI, 1974 | Zbl | Zbl

[47] Iwanjec H., “Rosser's sieve”, Acta arith., 36 (1980), 171–202 | MR

[48] Jurkat W. B., “Abstracts of short communications”, Proc. Internat. Congr. Math. (Stockholm, 1962), Inst. Mittag-Lefler, Djursholm, Sweden, 1963, 35

[49] Kuhn P., “Eine Verbesserung des Restiglieds beim elementaren Beweis des Prinzahlsatzes”, Math. Scand., 3 (1955), 75–89 | MR | Zbl

[50] Vallee Poussin C. J. de la., “Recherches analytiques sur la théorie des nombres premiers”, Ann. soc. sci. Bruxelles, 20 (1896), 183–256; Sur la fonctions $\zeta(s)$ de Riemann et le nombrs des nombers premiers inférieurs à une limite donnée, Memoirs couronnés de l'Acad. roy. des sciences. Belgique, 59, 1899–1900 ; reprinted in Colloque sur la théorie des nombres (Bruxelles, 1955), Thone, Liege, 1956, 9–66 | Zbl

[51] Landau E., Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig, 1909; reprinted (with an appendix by P. T. Bateman) Chelsea, New York, 1953 ; “Über einige neuere Grenzwertsätze”, Rend. circ. mat. Palermo, 34 (1912), 121–131 ; Vorlesungen über Zahlentheorie, Hirzel, Leipzig, 1927 ; reprinted by Chelsea, New York, 1974; “Über den Wienerschen neuen Weg zum Primzahlsatz”, Sitzber. Preuss. Akad. Wiss., 1932, 514–521 ; reprinted in Handbuch, 1953, 917–924 | MR | Zbl | DOI | Zbl | Zbl | Zbl

[52] Langevin M., “Méthodes élémentaires en vue du théorème de Sylvester”, Théorie des nombres: Fasc. 2 Exp. No. G2, Sem. Delange-Pisot-Poitou, 17e année, 1975/76, Secretariat Math., Paris, 1977 | MR

[53] Lavrik A. F., Sobirov A. Sh., “Ob ostatochnom chlene v elementarnom dokazatelstve teoremy o chisle prostykh chisel”, Dokl. AN SSSR, 211:3 (1973), 534–536 | MR | Zbl

[54] Legendre A. M., Essai sur la théorie des nombres, Duprat, Paris, 1798

[55] Lehman R. S., “On the difference $\pi(x)-\mathrm{li}(x)$”, Acta arith., 11 (1966), 397–410 | MR | Zbl

[56] Levin B. V., Fainleib A. S., “Primenenie nekotorykh integralnykh uravnenii k voprosam teorii chisel”, Uspekhi mat. nauk, 22:3 (1967), 119–197 | MR | Zbl

[57] Levinson N., “A motivated account of an elementary proof of the prime number theorem”, Amer. Math. Monthly, 76 (1969), 225–245 | DOI | MR | Zbl

[58] Littlewood J. E., “Sur la distribution des nombres premiers”, C. r. Acad. sci. Paris, 158 (1914), 1869–1872 | Zbl

[59] Mangoldt H. von, “Zu Riemanns Abhandlung, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse””, J. reine angew. Math., 114 (1895), 255–305; “Zur Verteilung der Nullstellen der Riemannschen Funktion $\zeta(t)$”, Math. Ann., 60 (1905), 1–19 | DOI | MR | Zbl

[60] Mathews G. B., Theory of numbers, I, Deighton Bell, Cambridge, England, 1892 ; reprinted by Stechert, New York, 1927 | Zbl

[61] Meissel E., “Ueber die Bestimmung der Primzahlmenge inner halb gegebener Grenzen”, Math. Ann., 2 (1870), 636–642 | DOI | MR

[62] Mertens F., “Ein Beitrag zur analytischen Zahlentheorie”, J. reine angew. Math., 78 (1874), 46–62

[63] Montgomery H. L., Topics in multiplicative number theory, Lecture Notes in Math., 227, Springer-Verlag, Berlin and New York, 1971 ; Montgomeri Kh. L., Multiplikativnaya teoriya chisel, Mir, M., 1974 | MR | Zbl | MR

[64] Nagell T., Introduction to number theory, Wiley, New York, 1951 | MR | Zbl

[65] Nair M., “On Chebyshev-type inequalities for primes”, Amer. Math. Monthly., 89 (1982), 126–129 | DOI | MR | Zbl

[66] Nevanlinna V., “Über die elementaren Beweise der Primzahlsatze und deren äquivalente Fassungen”, Ann. Acad. Sci. Fenn. Ser. AI, 1964, no. 343 | MR

[67] Pintz J., “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 (1877), 173–178 ; 32 (1977), 397–406 ; corrigendum, 33 (1977), 293–295 ; 33 (1977), 89–98 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl | MR | MR | Zbl | MR | Zbl | MR | Zbl

[68] Pitz H. R., Tauberian theorems, Oxford, London, 1958

[69] Polya G., “Heuristic reasoning in the theory of numbers”, Amer. Math. Monthly, 66 (1959), 375–384 | DOI | MR | Zbl

[70] Popken J., “On convolutions in number theory”, Indag. Math., 17 (1955), 10–15 | MR

[71] Postnikov A. G., Romanov N. P., “Uproschenie elementarnogo dokazatelstva A. Selberga asimptoticheskogo zakona raspredeleniya prostykh chisel”, Uspekhi mat. nauk, 10:4 (1955), 75–87 ; “Замечание в статье А. Г. Постникова и Н. П. Романова”, Успехи мат. наук, 24:5 (1969), 263 | MR | Zbl | MR | Zbl

[72] Prachar K., Prinzahlverteilung, Die Grundelehren der math. Wiss., Springer, Berlin, 1957 ; Ppaxap K., Raspredelenie prostykh chisel, Mir, M., 1967 | Zbl | MR

[73] Richert H. E., Lectures on sieve methods, Tata Inst., Bombay, 1976 | Zbl

[74] Riemann B., Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse, Monatsber Kgl. Preuss. Akad. Wiss., Berlin, 1860; also in Werke, 2nd ed., Teubner, Leipzig, 1892; reprinted by Dover, New York, 1953 | Zbl

[75] Ross P. M., “On Chen's theorem that each large even integer has the form $p_1+p_2$ or $p_1+p_2p_3$”, J. London Math. Soc. (2), 10 (1975), 500–506 | DOI | MR | Zbl

[76] Rosser J. B., Schoenfeld L., “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 6 (1962), 64–94 ; “Sharper bounds for the Chebyshev functions $\vartheta(x)$ and $\psi(x)$”, Math. Comp., 29 (1975), 243–269 ; “Sharper bounds for the Chebyshev functions $\vartheta(x)$ and $\psi(x)$, II”, Math. Comp., 30 (1976), 337–360 ; corrigendum, Math. Comp., 30 (1976), 900 | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI

[77] Schwarz W., Einführung in Methoden und Ergebnisse der Primzahltheorie, B.I-Hochschultaschenbücher, 278/278A, Bibliogr. Inst., Mannheim, 1969 | MR | Zbl

[78] Segal S. L., “Prime number theorem analogues without primes”, J. reine angew. Math., 265 (1974), 1–22 | MR | Zbl

[79] Selberg A., “An elementary proof of the prime number theorem”, Ann. of Math. (2), 50 (1949), 305–313 ; “An elementary proof of the prime number theorem for arithmetic progressions”, Canad. J. Math., 2 (1950), 66–78 ; “On elementary methods in prime number theory and their limitations”, Den 11te Skand. Mat. Kongress (Trondheim, 1949), Tanums Forlag, Oslo, 1952, 13–22 | DOI | MR | Zbl | MR | Zbl | MR

[80] Shapiro H. N., “On a theorem of Selberg and generalizations”, Ann. of math. (2), 51 (1950), 485–497 ; “On primes in arithmetic progression, II”, Ann. of Math. (2), 52 (1952), 231–243 | DOI | MR | Zbl | DOI | MR

[81] Sokolovski A. V., Letter to H. Diamond and J. Steinig, 17 February, 1973

[82] Specht W., Elementare Beweise der Primzahlsatze, Hochschulbucher, Bd. 30, VEB Deutcher Verlag Wiss., Berlin, 1956 | MR

[83] Sylvester J. J., “On Tchebycheff's theorem of the totality of prime numbers comprised within given limits”, Amer. J. Math., 4 (1881), 230–247 ; “On arithmetical series”, Messenger of Math. (2), 21 (1892), 1–19 and 87–120 | DOI | MR

[84] Tatuzawa T., Iseki K., “On Selberg's elementary proof of the prime number theorem”, Proc. Japan Acad., 27 (1951), 340–342 | DOI | MR | Zbl

[85] Titchmarsh E. C., The theory of the Riemann zeta function, Oxford, London, 1951 ; Titchmarsh E., Teoriya dzeta-funktsii Rimana, IL, M., 1953 | MR

[86] Vinogradov A. I., “O plotnostnoi gipoteze dlya $L$-ryadov Dirikhle”, Izv. AN SSSR. Ser. mat., 29:4 (1965), 903–934 | Zbl

[87] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1971 | MR

[88] Walfisz A., “Weylsche Exponentialsummen in der neueren Zahlentheorie”, Math. Forschungsber. XV, VEB Deutscher Verlag Wiss, Berlin, 1963 | MR | Zbl

[89] Widder D. V., The Laplace transform, Princeton Univ. Press, Princeton, NH, 1941 | MR | Zbl

[90] Wiener N., “A new method in Tauberian theorem”, Math. Phys. M.I.T., 7 (1927–28), 161–184 ; “Tauberian theorems”, Ann. of Math. (2), 33 (1932), 1–100 | MR | DOI | MR | Zbl

[91] Wiener N., Geller L., “Some prime number consequences of the Ikehara theorem”, Acta sci. math. Szeged., 12B (1950), 25–28 | MR

[92] Wintner A., The theory of measure in arithmetical semigroups, Waverly Press., Baltimore, 1944

[93] Wirsing E., “Elementare Beweise des Primzahlsatzes mit Restiglied. I”, J. reine angew. Math., 211 (1962), 205–214 ; “Elementare Beweise des Primzahlsatzes mit Restglied. II”, J. reine angew. Math., 214/215 (1963), 1–18 ; “Des asymptotische Verhalten von Summen über multiplikative Funktionen. II”, Acta math. acad. sci. Hungar., 18 (1967), 411–467 ; Virzing E., “Elementarnoe dokazatelstvo teoremy o prostykh chislakh s ostatochnym chlenom. I i II”, Sbor. per.: Matematika, 9:2 (1965), 45–54, 55–71 | MR | Zbl | MR | DOI | MR | Zbl

[94] Wright E. M., “The elementary proof of the prime number theorem”, Proc. Roy. Soc. Edinburgh, A63 (1952), 257–267 | MR

[95] Math. Intelligencer, 1977, 7–10 | DOI | MR | Zbl