# zbMATH — the first resource for mathematics

Goldbach’s problem. (Das Goldbach’sche Problem.) (German) Zbl 0805.11003
The author presents a historical survey on Goldbach’s problem, i.e. the representation of integers as sums of two or three primes. He sketches the Hardy-Littlewood circle method and the results obtained from it, I. M. Vinogradov’s solution of the ternary Goldbach problem, the implications of sieve methods, and more recent results (for example the estimate of Montgomery-Vaughan for the number of exceptional, non- Goldbach-integers), localized forms on the ternary problem, the deep Heath-Brown hypothetical result (prime twins and Siegel zeros) and others. Finally some open problems are mentioned.

##### MSC:
 11-03 History of number theory 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method 01-02 Research exposition (monographs, survey articles) pertaining to history and biography 11N35 Sieves 11M20 Real zeros of $$L(s, \chi)$$; results on $$L(1, \chi)$$
Full Text:
##### References:
 [1] Balog, A.: Linear equations in primes. Matematika39, 367–378 (1992) · Zbl 0774.11058 [2] Bombieri, E.: On the large sieve. Matematika12, 201–225 (1965) · Zbl 0136.33004 [3] Chen, J.-R.: The exceptional set of Goldbach numters, II. Sci. Sin. Ser. A26, 714–731 (1983) · Zbl 0513.10045 [4] Chen, J.-R., Ze, W.T.: On the Goldbach Problem. Acta Math. Sin.32, 702–718 (1989) · Zbl 0695.10041 [5] Corput, J.G. van der: Sur l’hypothèse de Goldbach pour presque tous les nombres premiers. Acta Arith.2, 266–290 (1937) · Zbl 0018.05203 [6] Davenport, H.: Multiplikative number theory. 2. ed. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0453.10002 [7] Descartes, R.: Opuscula Posthuma, Excerpta Mathematica. Oeuvres, Bd. X. Amsterdam, 1701 [8] Erdös, P.: Some results on additive number theory. Proc. Am. Math. Soc.5, 847–853 (1954) · Zbl 0056.27001 [9] Estermann, T.: On Goldbach’s problem: Proof that almost all even positive integers are sums of two primes. Proc. Lond. Soc. II44, 307–314 (1938) · Zbl 0020.10503 [10] Euler, L. Briefwechsel. Im Erscheinen. Nr. 765.766. Kommentiert in Bd. I. Basel: Birkhäuser 1957. Abgedruckt in Fuss, P.H.: Correspondence Mathématique et Physique. Tome 1. St. Petersburg 1843. Nachdruck durch Johnson Reprint Corp., New York – London,: Johnson 1968 [11] Gallagher, P.X.: A large sieve density estimate nears=1. Invent. Math.11, 329–339 (1970) · Zbl 0219.10048 [12] Halberstam, H., Richert, H.E.: Sieve methods. London, New York, San Francisco: Academic Press 1974 · Zbl 0298.10026 [13] Halberstam, H., Roth, K.F.: Sequences. Oxford: Clarendon 1966 [14] Hardy, G.H., Littlewood, J.E.: Some problems of ”Partitio Numerorum”, II. On the expression of a number as a sum of primes. Acta Math.44, 1–70 (1922) V. A further contribution to the study of Goldbach’s problem. Proc. Lond. Math Soc.22, 46–56 (1924) · JFM 48.0143.04 [15] Heath-Brown, D.R.: Prime twins and Siegel zeros. Proc. Lond. Math. Soc.47, 193–224 (1983) · Zbl 0517.10044 [16] Heilbronn, H.: Zbl. Math.16, 291–292 (1937) [17] Hilbert, D.: Mathematische Probleme. Vortrag, geh. auf dem Intern. Math. Kongr. Paris 1900. Göttinger Nachr. 1900, 253–297. Ges. Abh. III, Berlin: Springer 1935 · JFM 31.0068.03 [18] Hornfeck, B.: Ein Satz über die Primzahlmenge. Math. Z.60, 271–273 (1954) · Zbl 0056.03901 [19] Karatsuba, A.A.: Basic analytic number theory. Berlin, Heidelberg, New York: Springer 1983 [20] Landau, E.: Über die zahlentheoretische Funktion ({$$\eta$$}) und ihre Beziehung zum Goldbach’schen Satz. Gött. Nachr. 1900, 177–186 · JFM 31.0179.01 [21] Landau, E.: Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemann’schen Zetafunktion. Vortrag, geh. auf dem Internat. Math. Kongr. Cambridge, 1912. Jahresber. Deutsche Math.-Verein.21, 208–228 (1912) · JFM 43.0264.01 [22] Montgomery, H.L., Vaughan, R.C.: The exceptional set in Goldbach’s problem. Acta Arith.27, 253–270 (1975) · Zbl 0301.10043 [23] Perelli, A., Pintz, J.: On the exceptional set for Goldbach’s problem in short intervals. Compos. Math.82, 355–372 (1992) · Zbl 0756.11028 [24] Prachar, K.: Über die Lösungszahl eines Systems von Gleichungen in Primzahlen. Monatsch. Math.59, 98–103 (1955) · Zbl 0064.04108 [25] Prachar, K.: Primzahlverteilung. Berlin, Göttingen, Heidelberg: Springer 1957 [26] Ribenboim, P.: The book of prime number records. Berlin, Heidelberg, New York: Springer 1988 · Zbl 0642.10001 [27] Rieger, G.J.: Über ein lineares Gleichungsystem von Prachar mit Primzahlen. J. Reine Angew. Math.213, 103–107 (1963/64) · Zbl 0121.04901 [28] Riesel, H., Vaughan, R.C.: On sums of primes. Ark. Mat.21, 46–74 (1983) · Zbl 0516.10044 [29] Schwarz, W.: Einführung in die Methoden und Ergebnisse der Primzahltheorie. Mannheim: BI 1969 · Zbl 0217.31601 [30] Sylvester, J.J.: On the partition of an even number into two primes. Proc. Lond. Math. Soc. 4–6 (1871–73). Coll. Math. papers II New York: Chelsea 1973 [31] Tschudakov, N.G.: On the density of a set of even integers which are not representable as a sum of two odd primes (Russisch). Izv. Akad. Nauk SSSR, Ser. Mat.1, 25–40 (1938) [32] Vaughan, R.C.: Sommes trigonométriques sur les nombres premires. C.R. Acad. Sci. Paris Ser. A-B285, A981-A983 (1977) · Zbl 0374.10025 [33] Vaughan, R.C.: The Hardy-Littlewood Method. Cambridge Tracts in Mathematics, 80. Cambridge, New York: Cambridge University Press 1981 · Zbl 0455.10034 [34] Vinogradov, I.M.: Representation of an odd number as the sum of three primers. Dokl. Akad. Nauk SSSR15, 291–294 (1937) [35] Wirsing, E.: Thin subbases. Analysis6, 285–308 (1986) · Zbl 0586.10032 [36] Wolke, D.: Some applications of zero density theorems forL-functions. Acta Math. Hung.61, 241–258 (1993) · Zbl 0790.11075 [37] Zhan, T.: On the representation of large odd integers as the sum of three almost equal primes. Acta Math. Sin.7, 259–272 (1991) · Zbl 0742.11048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.