Landau’s problems on primes
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 357-404.

At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :

  • (1) Are there infinitely many primes of the form n 2 +1?
  • (2) The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.
  • (3) The Twin Prime Conjecture.
  • (4) Does there exist always at least one prime between neighbouring squares?

All these problems are still open. In the present work a survey will be given about partial results in Problems (2)–(4), with special emphasis on the recent results of D. Goldston, C. Yıldırım and the author on small gaps between primes.

Au congrès international de Cambridge en 1912, Laudau dressa la liste de quatre problèmes de base sur les nombres premiers. Ces problèmes furent caractérisés dans son discours comme “inaccessibles en l’état actuel de la science”. Ces problèmes sont les suivants :

  • (1) Existe-t-il une infinité de nombres premiers de la forme n 2 +1 ?
  • (2) La conjecture (binaire) de Goldbach, que chaque nombre pair supérieur à 2 est somme de deux nombres premiers.
  • (3) La conjecture des nombres premiers jumeaux.
  • (4) Existe-t-il toujours un nombre premier entre deux carrés consécutifs ?

Tous ces problèmes sont encore ouverts. Le travail présenté ici est un exposé des résultats partiels aux problèmes (2)–(4), avec une attention particuliere concernant les résultats récents de D. Goldston, C. Yıldırım et de l’auteur sur les petits écarts entre nombres premiers.

DOI: 10.5802/jtnb.676
Pintz, János 1

1 Rényi Mathematical Institute of the Hungarian Academy of Sciences Budapest Reáltanoda u. 13–15 H-1053, Hungary
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Pintz, János. Landau’s problems on primes. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 357-404. doi : 10.5802/jtnb.676. http://archive.numdam.org/articles/10.5802/jtnb.676/

[1] R. J. Backlund, Über die Differenzen zwischen den Zahlen, die zu den ersten n Primzahlen teilerfremd sind. Commentationes in honorem E. L. Lindelöf. Annales Acad. Sci. Fenn. 32 (1929), Nr. 2, 1–9.

[2] R. C. Baker, G. Harman, The difference between consecutive primes. Proc. London Math. Soc. (3) 72 (1996), 261–280. | MR | Zbl

[3] R. C. Baker, G. Harman, J. Pintz, The exceptional set for Goldbach’s problem in short intervals, in: Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995). 1–54, London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press, Cambridge, 1997. | MR | Zbl

[4] R. C. Baker, G. Harman, J. Pintz, The difference between consecutive primes, II. Proc. London Math. Soc. (3) 83 (2001), no. 3, 532–562. | MR | Zbl

[5] A. Balog, On the fractional part of p ϑ . Archiv Math. 40 (1983), 434–440. | MR | Zbl

[6] M. B. Barban, The density of zeros of Dirichlet L-series and the problem of the sum of primes and near primes. Mat. Sb. 61 (1963), 418–425 (Russian). | MR | Zbl

[7] P. T. Bateman, R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp. 16 (1962), 363–367. | MR | Zbl

[8] J. Bertrand, Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu’elle enferme. J. École Roy. Polytechnique 18 (1845), 123–140.

[9] E. Bombieri, On the large sieve. Mathematika 12 (1965), 201–225. | MR | Zbl

[10] E. Bombieri, H. Davenport, Small differences between prime numbers. Proc. Roy. Soc. Ser. A 293 (1966), 1–18. | MR | Zbl

[11] A. Brauer, H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre. Sber. Berliner Math. Ges. 29 (1930), 116–125. | JFM

[12] H. Brocard, L’intermédiaire des math. 4 (1897), p. 159.

[13] V. Brun, Le crible d’Eratosthéne et le théorème de Goldbach. Videnselsk. Skr. 1 (1920), Nr. 3. | JFM

[14] A. A. Buhštab, New improvements in the sieve of Eratosthenes. Mat. Sb. 4 (1938), 357–387 (Russian). | Zbl

[15] A. A. Buhštab, Sur la décomposition des nombres pairs en somme de deux composantes dont chacune est formée d’un nombre borné de facteurs premiers. Doklady Akad. Nauk. SSSR 29 (1940), 544–548. | JFM | MR

[16] A. A. Buhštab, New results in the investigation of Goldbach–Euler’s problem and the problem of twin prime numbers. Doklady Akad. Nauk. SSSR 162 (1965), 735–738 (Russian). | MR | Zbl

[17] A. A. Buhštab, A combinatorial strengthening of the Eratosthenian sieve method. Usp. Mat. Nauk 22 (1967), no. 3, 199–226 (Russian). | MR | Zbl

[18] Y. Buttkewitz, Master’s Thesis. Freiburg Univ., 2003.

[19] F. Carlson, Über die Nullstellen der Dirichletschen Reihen und der Riemannscher ζ-Funktion. Arkiv f. Math. Astr. Fys. 15 (1920), No. 20. | JFM

[20] P. L. Čebyšev, Mémoire sur les nombres premiers. Mémoire des seuvants étrangers de l’Acad. Sci. St. Pétersbourg 7 (1850), 17–33.

[21] Jing Run Chen, 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 (Chinese). | MR

[22] Jing Run Chen, 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

[23] Jing Run Chen, On the distribution of almost primes in an interval. Sci. Sinica 18 (1975), 611–627. | MR | Zbl

[24] Jing Run Chen, Jian Min Liu, The exceptional set of Goldbach numbers, III. Chinese Quart. J. Math. 4 (1989), 1–15. | MR

[25] J. G. van der Corput, Sur l’hypothése de Goldbach pour presque tous les nombres pairs. Acta Arith. 2 (1937), 266–290. | EuDML | Zbl

[26] H. Cramér, Some theorems concerning prime numbers. Arkiv f. Math. Astr. Fys. 15 (1920), No. 5, 1–33. | JFM

[27] H. Cramér, Prime numbers and probability. Skand. Math. Kongr. 8 (1935), 107–115. | Zbl

[28] H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2 (1936), 23–46. | EuDML | Zbl

[29] N. G. Čudakov, On the zeros of Dirichlet’s L-functions. Mat. Sb. 1 (1936a), 591–602. | EuDML | JFM | Zbl

[30] N. G. Čudakov, On the difference between two neighbouring prime numbers. Mat. Sb. 1 (1936b), 799–814. | JFM | Zbl

[31] N. G. Čudakov, On the density of the set of even numbers which are not representable as a sum of two primes. Izv. Akad. Nauk. SSSR 2 (1938), 25–40. | JFM | Zbl

[32] H. Davenport, Multiplicative Number Theory. Revised by Hugh L. Montgomery, 2nd edition, Springer, Berlin, Heidelberg, New York, 1980. | MR | Zbl

[33] A. Desboves, Sur un théorème de Legendre et son application à la recherche de limites qui comprennent entre elles des nombres premiers. Nouv. Ann. Math. 14 (1855), 281–295. | EuDML

[34] Descartes, Opuscula Posthuma, Excerpta Mathematica. Vol. 10, 1908.

[35] J.-M. Deshouillers, Amélioration de la constante de Šnirelman dans le probléme de Goldbach. Sém. Delange, Pisot, Poitou 14 (1972/73), exp. 17. | EuDML | Numdam | MR | Zbl

[36] J.-M. Deshouillers, Sur la constante de Šnirelman. Sém. Delange, Pisot, Poitu 17 (1975/76), exp. 16. | EuDML | Numdam | Zbl

[37] J.-M. Deshouillers, H. Iwaniec, On the greatest prime factor of n 2 +1. Ann. Inst. Fourier 32 (1982), 1–11. | EuDML | Numdam | MR | Zbl

[38] J.-M. Deshouillers, G. Effinger, H. te Riele, D. Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis. Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 99–104. | EuDML | MR | Zbl

[39] L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers. Messenger of Math. (2), 33 (1904), 155–161. | JFM

[40] P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandl. Kgl. Preuß. Akad. Wiss. (1837), 45–81. [Werke, I, 313–342. G. Reimer, Berlin, 1889; French translation: J. math. pures appl. 4, 1839, 393–422.]

[41] P. D. T. A. Elliott, H. Halberstam, A conjecture in prime number theory. Symposia Mathematica 4 INDAM, Rome, 59–72, Academic Press, London, 1968/69. | MR | Zbl

[42] P. Erdős, On the difference of consecutive primes. Quart. J. Math. Oxford ser. 6 (1935), 124–128. | JFM | MR | Zbl

[43] P. Erdős, The difference of consecutive primes. Duke Math. J. 6 (1940), 438–441. | JFM | MR

[44] P. Erdős, Some problems on number theory, in: Analytic and elementary number theory (Marseille, 1983). Publ. Math. Orsay, 86-1 (1983), 53–57. | MR

[45] P. Erdős, L. Mirsky, The distribution of values of the divisor function d(n). Proc. London Math. Soc. (3) 2 (1952), 257–271. | MR | Zbl

[46] T. Estermann, Eine neue Darstellung und neue Anwendungen der Viggo Brunschen Methode. J. Reine Angew. Math. 168 (1932), 106–116. | EuDML | MR | Zbl

[47] T. Estermann, On Goldbach’s problem: Proof that almost all even positive integers are sums of two primes. Proc. London Math. Soc. (2) 44 (1938), 307–314. | JFM | MR

[48] É. Fouvry, F. Grupp, On the switching principle in sieve theory. J. Reine Angew. Math. 370 (1986), 101–126. | EuDML | MR | Zbl

[49] P.-H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIéme siécle. St. Pétersbourg, 1843. [Reprint: Johnson Reprint Co. 1968.] | MR | Zbl

[50] P. X. Gallagher, A large sieve density estimate near σ=1. Invent. Math. 11 (1970), 329–339. | EuDML | MR | Zbl

[51] P. X. Gallagher, Primes and powers of 2. Invent. Math. 29 (1975), 125–142. | EuDML | MR | Zbl

[52] D. A. Goldston, On Hardy and Littlewood’s contribution to the Goldbach conjecture, in: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989). 115–155, Univ. Salerno, Salerno, 1992. | MR | Zbl

[53] D. A. Goldston, Y. Motohashi, J. Pintz, C. Y. Yıldırım, Small gaps between primes exist. Proc. Japan Acad. 82A (2006), 61–65. | MR | Zbl

[54] D. A. Goldston, J. Pintz, C. Yıldırım, Primes in Tuples. Annals of Math. (200?), to appear, AIM Preprint Series, No. 2005-19, http://aimath.org/preprints.html | Zbl

[55] D. A. Goldston, J. Pintz, C. Yıldırım, Primes in Tuples II. Acta Math. (200??), to appear, preprint at arXiv:0710.2728 | MR | Zbl

[56] D. A. Goldston, J. Pintz, C. Yıldırım, Primes in Tuples III: On the difference p n+ν -p n . Funct. Approx. Comment. Math. 35 (2006), 79–89. | MR | Zbl

[57] D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yıldırım, Small gaps between products of two primes. Proc. London Math. Soc. (200?), to appear, preprint at arXiv:math/0609615 | Zbl

[58] D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdős. (200??), Preprint at arXiv: 0803.2636

[59] A. Granville, Unexpected irregularities in the distribution of prime numbers, in: Proceedings of the International Congress of Mathematicians (Zürich, 1994). Vol. 1, 2, 388–399, Birkhäuser, Basel, 1995. | MR | Zbl

[60] A. Granville, Harald Cramér and the Distribution of Prime Numbers. Scand. Actuarial J. No. 1 (1995), 12–28. | MR | Zbl

[61] G. Greaves, Sieves in Number Theory. Springer, 2001. | MR | Zbl

[62] J. Hadamard, Sur les zéros de la fonction ζ(s) de Riemann. Comptes Rendus Acad. Sci. Paris 122 (1896), 1470–1473. | JFM

[63] J. Hadamard, Sur la fonction ζ(s). Comptes Rendus Acad. Sci. Paris 123 (1896), p. 93. | JFM

[64] J. Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquances arithmétiques. Bull. Soc. Math. France 24 (1896), 199–220. | EuDML | JFM | Numdam | MR

[65] G. H. Hardy, J. E. Littlewood, Some problems of ‘Partitio Numerorum’, III: On the expression of a number as a sum of primes. Acta Math. 44 (1923), 1–70. | JFM | MR

[66] G. H. Hardy, J. E. Littlewood, Some problems of ’Partitio Numerorum’, V: A further contribution to the study of Goldbach’s problem. Proc. London Math. Soc. (2) 22 (1924), 46–56. | JFM | MR

[67] G. Harman, Primes in short intervals. Math. Zeitschr. 180 (1982), 335–348. | EuDML | MR | Zbl

[68] G. Harman, On the distribution of p modulo one. Mathematika 30 (1983), 104–116. | MR | Zbl

[69] D. R. Heath-Brown, A parity problem from sieve theory. Mathematika 29 (1982), 1–6. | MR | Zbl

[70] D. R. Heath-Brown, Prime twins and Siegel zeros. Proc. London Math. Soc. (3) 47 (1983), 193–224. | MR | Zbl

[71] D. R. Heath-Brown, The divisor function at consecutive integers. Mathematika 31 (1984), 141–149. | MR | Zbl

[72] D. R. Heath-Brown, Almost-prime k-tuples. Mathematika 44 (1997), 245–266. | MR | Zbl

[73] D. R. Heath-Brown, H. Iwaniec, On the difference between consecutive prime numbers. Invent. Math. 55 (1979), 49–69. | EuDML | MR | Zbl

[74] D. R. Heath-Brown, J.-C. Puchta, Integers represented as a sum of primes and powers of two. Asian J. Math. 6 (2002), no. 3, 535–565. | MR | Zbl

[75] H. Heilbronn, E. Landau, P. Scherk, Alle großen ganzen Zahlen lassen sich als Summe von höchstens 71 Primzahlen darstellen. Časopis pěst. Math. Fys. 65 (1936), 117–141. [E. Landau, Collected Works, 9, 351–375, Thales Verlag; The Collected Papers of Hans Arnold Heilbronn, 197–211, J. Wiley 1988.] | EuDML | JFM | Zbl

[76] A. J. Hildebrand, Erdős’ problems on consecutive integers, Paul Erdős and his Mathematics I. Bolyai Society Mathematical Studies 11, Budapest, 2002, 305–317. | MR | Zbl

[77] D. Hilbert, Gesammelte Abhandlungen. Vol. 3, 290–329, Springer, Berlin, 1935. | Zbl

[78] G. Hoheisel, Primzahlprobleme in der Analysis. SBer. Preuss. Akad. Wiss., Berlin, 1930, 580–588. | JFM

[79] C. Hooley, On the greatest prime factor of a quadratic polynomial. Acta Math. 117 (1967), 281–299. | MR | Zbl

[80] M. N. Huxley, On the differences of primes in arithmetical progressions. Acta Arith. 15 (1968/69), 367–392. | EuDML | MR | Zbl

[81] M. N. Huxley, On the difference between consecutive primes. Invent math. 15 (1972), 164–170. | EuDML | MR | Zbl

[82] M. N. Huxley, Small differences between consecutive primes. Mathematika 20 (1973), 229–232. | MR | Zbl

[83] M. N. Huxley, Small differences between consecutive primes II. Mathematika 24 (1977), 142–152. | MR | Zbl

[84] M. N. Huxley, An application of the Fouvry–Iwaniec theorem. Acta Arith. 43 (1984), 441–443. | EuDML | MR | Zbl

[85] A. E. Ingham, On the difference between consecutive primes. Quart. J. Math. Oxford ser. 8 (1937), 255–266. | JFM

[86] H. Iwaniec, Almost primes represented by quadratic polynomials. Invent. math. 47 (1978), 171–188. | EuDML | MR | Zbl

[87] H. Iwaniec, M. Jutila, Primes in short intervals. Arkiv Mat. 17 (1979), 167–176. | MR | Zbl

[88] H. Iwaniec, J. Pintz, Primes in short intervals. Monatsh. Math. 98 (1984), 115–143. | EuDML | MR | Zbl

[89] Chaohua Jia, Difference between consecutive primes. Sci. China Ser. A 38 (1995a), 1163–1186. | MR | Zbl

[90] Chaohua Jia, Goldbach numbers in a short interval, I. Science in China 38 (1995b), 385–406. | MR | Zbl

[91] Chaohua Jia, Goldbach numbers in a short interval, II. Science in China 38 (1995c), 513–523. | MR | Zbl

[92] Chaohua Jia, Almost all short intervals containing prime numbers. Acta Arith. 76 (1996a), 21–84. | EuDML | MR | Zbl

[93] Chaohua Jia, On the exceptional set of Goldbach numbers in a short interval. Acta Arith. 77 (1996b), no. 3, 207–287. | EuDML | MR | Zbl

[94] Chaohua Jia, Ming-Chit Liu, On the largest prime factor of integers. Acta Arith. 95 (2000), No. 1, 17–48. | EuDML | MR | Zbl

[95] M. Jutila, On numbers with a large prime factor, I. J. Indian Math. Soc. (N.S.) 37 (1973), 43–53. | MR | Zbl

[96] M. Jutila, On Linnik’s constant. Math. Scand. 41 (1977), 45–62. | EuDML | MR | Zbl

[97] L. Kaniecki, On Šnirelman’s constant under the Riemann hypothesis. Acta Arith. 72 (1995), 361–374. | EuDML | MR | Zbl

[98] J. Kaczorowski, A. Perelli, J. Pintz, A note on the exceptional set for Goldbach’s problem in short intervals. Monatsh. Math. 116, no. 3-4 (1995), 275–282. Corrigendum: ibid. 119 (1995), 215–216. | EuDML | MR | Zbl

[99] I. Kátai, A remark on a paper of Ju. V. Linnik. Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 17 (1967), 99–100. | MR | Zbl

[100] R. M. Kaufman, The distribution of {p}. Mat. Zametki 26 (1979), 497–504 (Russian). | MR | Zbl

[101] N. I. Klimov, On the computation of Šnirelman’s constant. Volžskij Mat. Sbornik 7 (1969), 32–40 (Russian). | MR

[102] N. I. Klimov, A refinement of the estimate of the absolute constant in the Goldbach–Šnirelman problem, in: Number theory: Collection of Studies in Additive Number Theory. Naučn. Trudy Kuibyšev. Gos. Ped. Inst. 158 (1975), 14–30 (Russian). | MR

[103] N. I. Klimov, A new estimate of the absolute constant in the Goldbach–Šnirelman problem. Izv. VUZ no. 1 (1978), 25–35 (Russian). | MR | Zbl

[104] N. I. Klimov, G. Z. Pilt’ai, T. A. Šeptickaja, An estimate for the absolute constant in the Goldbach–Šnirelman problem. Issled. po teorii čisel, Saratov No. 4 (1972), 35–51 (Russian). | MR | Zbl

[105] S. Knapowski, On the mean values of certain functions in prime number theory. Acta Math. Acad. Sci. Hungar. 10 (1959), 375–390. | MR | Zbl

[106] H. von Koch, Sur la distribution des nombres premiers. Acta Math. 24 (1901), 159–182. | JFM | MR

[107] L. Kronecker, Vorlesungen über Zahlentheorie, I. p. 68, Teubner, Leipzig, 1901. | Zbl

[108] P. Kuhn, Zur Vigo Brun’schen Siebmethode I. Norske Vid. Selsk. 14 (1941), 145–148. | JFM | MR | Zbl

[109] P. Kuhn, Neue Abschätzungen auf Grund der Viggo Brunschen Siebmethode. 12. Skand. Mat. Kongr. (1953), 160–168. | MR | Zbl

[110] P. Kuhn, Über die Primteiler eines Polynoms. Proc. ICM Amsterdam 2 (1954), 35–37.

[111] E. Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. Jahresber. Deutsche Math. Ver. 21 (1912), 208–228. [Proc. 5th Internat. Congress of Math., I, 93–108, Cambridge 1913; Collected Works, 5, 240–255, Thales Verlag.] | EuDML | JFM

[112] A. Languasco, On the exceptional set of Goldbach’s problem in short intervals. Mh. Math. 141 (2004), 147–169. | MR | Zbl

[113] A. Languasco, J. Pintz, A. Zaccagnini, On the sum of two primes and k powers of two. Bull. London Math. Soc. 39 (2007), no. 5,  771–780. | MR | Zbl

[114] Hong Ze Li, Goldbach numbers in short intervals. Science in China 38 (1995), 641–652. | MR | Zbl

[115] Hong Ze Li, Primes in short intervals. Math. Proc. Cambridge Philos. Soc. 122 (1997), 193–205. | MR | Zbl

[116] Hong Ze Li, The exceptional set of Goldbach numbers. Quart. J. Math. Oxford Ser. (2) 50, no. 200 (1999), 471–482. | MR | Zbl

[117] Hong Ze Li, The exceptional set of Goldbach numbers, II. Acta Arith. 92, no. 1 (2000a), 71–88. | EuDML | MR | Zbl

[118] Hong Ze Li, The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes. Acta Arith. 92 (2000b), 229–237. | EuDML | MR | Zbl

[119] Hong Ze Li, The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes, II. Acta Arith. 96 (2001), 369–379. | EuDML | MR | Zbl

[120] Yu. V. Linnik, Prime numbers and powers of two. Trudy Mat. Inst. Steklov. 38 (1951), 152–169 (Russian). | MR | Zbl

[121] Yu. V. Linnik, Some conditional theorems concerning the binary Goldbach problem. Izv. Akad. Nauk. SSSR 16 (1952), 503–520. | MR | Zbl

[122] Yu. V. Linnik, Addition of prime numbers and powers of one and the same number. Mat. Sb. (N.S.) 32 (1953), 3–60 (Russian). | MR | Zbl

[123] J. Y. Liu, M. C. Liu, T. Z. Wang, The number of powers of 2 in a representation of large even integers, I. Sci. China Ser. A 41 (1998a), 386–397. | MR | Zbl

[124] J. Y. Liu, M. C. Liu, T. Z. Wang, The number of powers of 2 in a representation of large even integers, II. Sci. China Ser. A 41 (1998b), 1255–1271. | MR | Zbl

[125] J. Y. Liu, M. C. Liu, T. Z. Wang, On the almost Goldbach problem of Linnik. J. Théor. Nombres Bordeaux 11 (1999), 133–147. | EuDML | Numdam | MR | Zbl

[126] Hong-Quan Liu, Jie Wu, Numbers with a large prime factor. Acta Arith. 89, no. 2 (1999), 163–187. | EuDML | MR | Zbl

[127] S. T. Lou, Q. Yao, A Chebychev’s type of prime number theorem in a short interval, II. Hardy–Ramanujan J. 15 (1992), 1–33. | MR | Zbl

[128] S. T. Lou, Q. Yao, The number of primes in a short interval. Hardy–Ramanujan J. 16 (1993), 21–43. | MR | Zbl

[129] H. Maier, Small differences between prime numbers. Michigan Math. J. 35 (1988), 323–344. | MR | Zbl

[130] H. Maier, C. Pomerance, Unusually large gaps between consecutive primes. Trans. Amer. Math. Soc. 322 (1990), 201–237. | MR | Zbl

[131] E. Maillet, L’intermédiaire des math. 12 (1905), p. 108.

[132] H. Mikawa, On the exceptional set in Goldbach’s problem. Tsukuba J. Math. 16 (1992), 513–543. | MR | Zbl

[133] H. Mikawa, On the intervals between consecutive numbers that are sums of two primes. Tsukuba J. Math. 17, No. 2 (1993), 443–453. | MR | Zbl

[134] H. L. Montgomery, Zeros of L-functions. Invent. math. 8 (1969), 346–354. | EuDML | MR | Zbl

[135] H. L. Montgomery, R. C. Vaughan, The exceptional set in Goldbach’s problem. Acta Arith. 27 (1975), 353–370. | EuDML | MR | Zbl

[136] C. J. Mozzochi, On the difference between consecutive primes. J. Number Th. 24 (1986), 181–187. | MR | Zbl

[137] W. Narkiewicz, The Development of Prime Number Theory. From Euclid to Hardy and Littlewood. Springer, 2000. | MR | Zbl

[138] Cheng Dong Pan On the representation of even numbers as the sum of a prime and a near prime. Sci. Sinica 11 (1962), 873–888 (Russian). | MR

[139] Cheng Dong Pan On the representation of even numbers as the sum of a prime and a product of not more than 4 primes. Sci. Sinica 12 (1963), 455–473. | MR

[140] A. Perelli, J. Pintz, On the exceptional set for the 2k-twin primes problem. Compositio Math. 82, no. 3 (1992), 355–372. | EuDML | Numdam | MR | Zbl

[141] A. Perelli, J. Pintz, On the exceptional set for Goldbach’s problem in short intervals. J. London Math. Soc. (2) 47 (1993), 41–49. | MR | Zbl

[142] G. Z. Pilt’ai, On the size of the difference between consecutive primes. Issledovania po teorii chisel, 4 (1972), 73–79. | MR | Zbl

[143] Ch. G. Pinner, Repeated values of the divisor function. Quart. J. Math. Oxford Ser. (2) 48, no. 192 (1997), 499–502. | MR | Zbl

[144] J. Pintz, On the remainder term of the prime number formula I. On a problem of Littlewood. Acta Arith. 36 (1980a), 341–365. | EuDML | MR | Zbl

[145] J. Pintz, On the remainder term of the prime number formula V. Effective mean value theorems. Studia Sci. Math. Hungar. 15 (1980b), 215–223. | MR | Zbl

[146] J. Pintz, On the remainder term of the prime number formula VI. Ineffective mean value theorems. Studia Sci. Math. Hungar. 15 (1980c), 225–230. | MR | Zbl

[147] J. Pintz, On primes in short intervals, I. Studia Sci. Math. Hungar. 16 (1981), 395–414. | MR | Zbl

[148] J. Pintz, On primes in short intervals, II. Studia Sci. Math. Hungar. 19 (1984), 89–96. | MR | Zbl

[149] J. Pintz, Very large gaps between consecutive primes. J. Number Th. 63 (1997), 286–301. | MR | Zbl

[150] J. Pintz, Recent Results on the Goldbach Conjecture, in: Elementare und Analytische Zahlentheorie (Tagungsband). Proceedings ELAZ-Conference, May 24–28, 2004, Steiner Verlag, Stuttgart, 2006, pp. 220–254. | MR | Zbl

[151] J. Pintz, Cramér vs. Cramér. On Cramér’s probabilistic model for primes. Funct. Approx. Comment. Math. 37 (2007), part 2, 361–376. | MR | Zbl

[152] J. Pintz, I. Z. Ruzsa, On Linnik’s approximation to Goldbach’s problem, I. Acta Arith. 109 (2003), no. 2, 169–194. | EuDML | MR | Zbl

[153] J. Pintz, I. Z. Ruzsa, On Linnik’s approximation to Goldbach’s problem, II. Manuscript (200?).

[154] A. de Polignac, Six propositions arithmologiques sur les nombres premiers. Nouv. Ann. Math. 8 (1849), 423–429. | EuDML | Numdam

[155] G. Pólya, Heuristic reasoning in the theory of numbers. Amer. Math. Monthly 66 (1959), 375–384. | MR | Zbl

[156] H. Rademacher, Beiträge zur Viggo Brunschen Methode in der Zahlentheorie. Abh. Math. Sem. Hamburg 3, 12–30. [Collected Papers 1 (1924), 259–277, MIT Press, 1974.] | JFM

[157] K. Ramachandra, A note on numbers with a large prime factor. J. London Math. Soc. (2) 1 (1969), 303–306. | MR | Zbl

[158] K. Ramachandra, On the number of Goldbach numbers in small intervals. J. Indian Math. Soc. (N.S.) 37 (1973), 157–170. | MR | Zbl

[159] O. Ramaré, On Šnirel’man’s constant. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) 22 (1995), 645–706. | EuDML | Numdam | MR | Zbl

[160] R. A. Rankin, The difference between consecutive prime numbers. J. London Math. Soc. 13 (1938), 242–244. | JFM

[161] R. A. Rankin, The difference between consecutive prime numbers. II. Proc. Cambridge Philos. Soc. 36 (1940), 255–266. | JFM | MR

[162] R. A. Rankin, The difference between consecutive prime numbers, V. Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 331–332. | MR | Zbl

[163] A. Rényi, On the representation of an even number as the sum of a single prime and a single almost-prime number. Doklady Akad. Nauk SSSR 56 (1947), 455–458 (Russian). | MR | Zbl

[164] A. Rényi, On the representation of an even number as the sum of a single prime and a single almost-prime number. Izv. Akad. Nauk SSSR 12 (1948), 57–78 (Russian). | MR | Zbl

[165] Sz. Gy. Révész, Effective oscillation theorems for a general class of real-valued remainder terms. Acta Arith. 49 (1988), 481–505. | EuDML | MR | Zbl

[166] G. Ricci, Su la congettura di Goldbach e la costante di Schnirelmann. Boll. Un. Math. Ital. 15 (1936), 183–187. | Zbl

[167] G. Ricci, Su la congettura di Goldbach e la costante di Schnirelmann, II, III. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (2) 6 (1937), 71–90, 91–116. | EuDML | JFM | Numdam | MR

[168] G. Ricci, La differenza di numeri primi consecutivi. Rendiconti Sem. Mat. Univ. e Politecnico Torino 11, 149–200. Corr. ibidem 12 (1952), p. 315. | MR | Zbl

[169] G. Ricci, Sull’andamento della differenza di numeri primi consecutivi. Riv. Mat. Univ. Parma 5 (1954), 3–54. | MR | Zbl

[170] H. Riesel, R. C. Vaughan, On sums of primes. Arkiv Mat. 21 (1983), 45–74. | MR | Zbl

[171] L. Ripert, L’Intermédiaire des Math. 10 (1903), p. 66.

[172] N. P. Romanov, On Goldbach’s problem. Tomsk, Izv. Mat. Tek. I (1935),34–38 (Russian). | Zbl

[173] A. A. Šanin, Determination of constants in the method of Brun–Šnirelman. Volzh. Mat. Sb. 2 (1964), 261–265 (Russian). | MR | Zbl

[174] Y. Saouter, Checking the odd Goldbach conjecture up to 10 20 . Math. Comp. 67 (1998), 863–866. | MR | Zbl

[175] A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4 (1958), 185–208; Erratum: ibidem 5 p. 259. Erratum: ibidem 5, p. 259. | EuDML | MR | Zbl

[176] J.-C. Schlage-Puchta, The equation ω(n)=ω(n+1). Mathematika 50 (2003), no. 1-2, 99–101 (2005). | MR | Zbl

[177] L. G. Schnirelman, On additive properties of numbers. Izv. Donsk. Politehn. Inst. 14 (1930), 3–28 (Russian).

[178] L. G. Schnirelman, Über additive Eigenschaften von Zahlen. Math. Ann. 107 (1933), 649–690. | EuDML | MR | Zbl

[179] A. Selberg, The general sieve method and its place in prime number theory. Proc. Internat. Congress of Math., Cambridge, Mass. 1 (1950), 286–292. | MR | Zbl

[180] A. Selberg, Collected Papers, Vol. II. Springer, Berlin, 1991. | MR | Zbl

[181] C. L. Siegel, Über die Klassenzahl quadratischer Körper. Acta Arith. 1, 83–86. [Gesammelte Abhandlungen 1 (1936), 406–409, Springer, Berlin–Heidelberg, 1966.] | EuDML | JFM | Zbl

[182] C. Spiro, Thesis. Urbana, 1981.

[183] J. J. Sylvester, On the partition of an even number into two primes. Proc. London Math. Soc. 4 (1871), 4–6. [Collected Math. Papers, 2, 709–711, Cambridge, 1908.] | JFM

[184] W. Tartakowski, Sur quelques sommes du type de Viggo Brun. C. R. Acad. Sci. URSS, N.S. 23 (1939a), 121–125. | JFM

[185] W. Tartakowski, La méthode du crible approximatif “électif”. C. R. Acad. Sci. URSS, N. S. 23 (1939b), 126–129. | JFM

[186] P. Turán, On the remainder term of the prime number formula I. Acta Math. Hungar. 1 (1950), 48–63. | MR | Zbl

[187] S. Uchiyama, On the difference between consecutive prime numbers. Acta Arith. 27 (1975), 153–157. | EuDML | MR | Zbl

[188] C. J. de la Vallée-Poussin, Recherches analytiques sur la théorie des nombres premiers, I–III. Ann. Soc. Sci. Bruxelles 20 (1896), 183–256, 281–362, 363–397. | JFM

[189] C. J. de la Vallée-Poussin, Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée. Mem Couronnés de l’Acad. Roy. Sci. Bruxelles, 59 (1899). | JFM

[190] R. C. Vaughan, On Goldbach’s problem. Acta Arith. 22 (1972), 21–48. | EuDML | MR | Zbl

[191] R. C. Vaughan, On the estimation of Schnirelman’s constant. J. Reine Angew. Math. 290 (1977), 93–108. | EuDML | MR | Zbl

[192] I. M. Vinogradov, Representation of an odd number as a sum of three prime numbers. Doklady Akad. Nauk. SSSR 15 (1937), 291–294 (Russian). | JFM

[193] I. M. Vinogradov, A certain general property of the distribution of prime numbers. Mat. Sb. 7 (1940), 365–372 (Russian). | JFM | MR | Zbl

[194] A. I. Vinogradov, Application of Riemann’s ζ(s) to the Eratosthenian sieve. Mat. Sb. 41 (1957), 49–80; Corr.: ibidem, 415–416 (Russian). | MR | Zbl

[195] A. I. Vinogradov, The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk. SSSR 29 (1965), 903–934 (Russian). Corr.: ibidem, 30 (1966), 719–720. | MR | Zbl

[196] I. M. Vinogradov, Special Variants of the Method of Trigonometric Sums. Nauka, Moskva, 1976 (Russian). | MR | Zbl

[197] Tianze Wang, On Linnik’s almost Goldbach theorem. Sci. China Ser. A 42 (1999), 1155–1172. | MR | Zbl

[198] Yuan Wang, On the representation of a large even integer as a sum of a product of at most 3 primes and a product of at most 4 primes. Acta Math. Sin. 6 (1956), 500–513 (Chinese). | MR | Zbl

[199] Yuan Wang, On sieve methods and some of their applications, I. Acta Math. Sinica 8 (1958), 413–429 (Chinese). [English translation: Sci. Sinica 8 (1959), 357–381.] | MR | Zbl

[200] Yuan Wang, Sheng-gang Xie, Kunrin Yu, Remarks on the difference of consecutive primes. Sci. Sinica 14 (1965), 786–788. | MR | Zbl

[201] E. Waring, Meditationes Algebraicae. Cantabrigine. 1770. [3rd. ed. 1782; English translation: American Math. Soc., Providence, 1991.] | MR

[202] N. Watt, Short intervals almost all containing primes. Acta Arith. 72 (1995), 131–167. | EuDML | MR | Zbl

[203] E. Westzynthius, Über die Verteilung der Zahlen, die zu der n ersten Primzahlen teilerfremd sind. Comm. Phys. Math. Helsingfors (5) 25 (1931), 1–37. | JFM | Zbl

[204] M. Y. Zhang, P. Ding, An improvement to the Schnirelman constant. J. China Univ. Sci. Techn. 13 (1983), Math. Issue, 31–53. | MR

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