A survey of some important problems in additive number theory

Vaughan, R. C.

Journées Arithmétiques de Luminy, Astérisque, no. 61 (1979), pp. 213-222.

MR Zbl

@incollection{AST_1979__61__213_0,
     author = {Vaughan, R. C.},
     title = {A survey of some important problems in additive number theory},
     booktitle = {Journ\'ees Arithm\'etiques de Luminy},
     series = {Ast\'erisque},
     pages = {213--222},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {61},
     year = {1979},
     mrnumber = {556674},
     zbl = {0401.10054},
     language = {en},
     url = {http://archive.numdam.org/item/AST_1979__61__213_0/}
}

TY  - CHAP
AU  - Vaughan, R. C.
TI  - A survey of some important problems in additive number theory
BT  - Journées Arithmétiques de Luminy
AU  - Collectif
T3  - Astérisque
PY  - 1979
SP  - 213
EP  - 222
IS  - 61
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_1979__61__213_0/
LA  - en
ID  - AST_1979__61__213_0
ER  -

%0 Book Section
%A Vaughan, R. C.
%T A survey of some important problems in additive number theory
%B Journées Arithmétiques de Luminy
%A Collectif
%S Astérisque
%D 1979
%P 213-222
%N 61
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_1979__61__213_0/
%G en
%F AST_1979__61__213_0

Vaughan, R. C. A survey of some important problems in additive number theory, dans Journées Arithmétiques de Luminy, Astérisque, no. 61 (1979), pp. 213-222. http://archive.numdam.org/item/AST_1979__61__213_0/

Bibliographie
Cité par

[1] E. F. Chechuro and A. A. Kuzjashev, The representation of large integers by sums of primes, Studies in Number Theory, n° 3, 45-50, Izdat. Saratov Univ., Saratov1969 | MR | Zbl

[2] J. R. Chen, On Waring's problem for $n$ -th powers, Acta Math. Sinica, 8 (1958), 253-257 ; | MR | Zbl

[2] J. R. Chen, On Waring's problem for $n$ -th powers, Chinese Math.-Acta 8 (1966), 849-853 (1967) | MR | Zbl

[3] J. R. Chen, Waring's problem for $g (5)$ , Sci. Record (Peking) (N.S.) 3 (1959), 327-330 | MR | Zbl

[4] J. R. Chen, Waring's problem for $g (5) = 37$ , Sci. Sinica 13 (1964), 335 | MR | Zbl

[5] J. R. Chen, Waring's problem for $g (5) = 37$ , Sci. Sinica 13 (1964), 1547-1568 | MR | Zbl

[6] J. R. 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 (Foreign Lang. Ed.) 17 (1966), 385-386 | MR

[7] J. R. 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 | Zbl

[8] N. G. Chudakov, On the density of the set of even numbers which are not representable as a sum of two odd primes, Izv. Akad. Nauk SSSR Ser.Nat. 2 (1938), 25-40 | JFM | Zbl

[9] N. G. Chudakov, On Goldbach-Vinogradov's theorem, Ann. of Math. (2) 48 (1947), 515-545 | DOI | MR | Zbl

[10] R. J. Cook, Pairs of additive equations, Michigan Math. J. 19 (1972) , 325-331 | DOI | MR | Zbl

[11] J. G. Van Der Corput, Sur l'hypothese de Goldbach pour presque tous les nombres pairs, Acta Arithmetica 2 (1937), 266-290 | DOI | EuDML | JFM | Zbl

[12]H. Davenport, On Waring's problem for fourth powers, Ann. of Math. 40 (1939), 731-747 | DOI | JFM | MR | Zbl

[13] H. Davenport, On sums of positive integeral $k$ -th powers, Amer. J. Math. 64 (1942), 189-198 | DOI | MR | Zbl

[14] H. Davenport, On Waring's problem for fifth and sixth powers, Amer. J. Math. 64 (1942) , 199-207 | DOI | MR | Zbl

[15] H. Davenport and D. J. Lewis, Homogeneous additive equations, Proc.Roy. Soc. Ser. A 274 (1963), 443-460 | DOI | MR | Zbl

[16]H. Davenport and D. J. Lewis, Cubic equations of additive type, Philos.Trans. Roy. Soc. Ser. A 261 (1966), 97-136 | DOI | MR | Zbl

[17]H. Davenport and D. J. Lewis, Simultaneous equations of additive type, Philos. Trans. Roy. Soc. Ser. A 264 (1969), 557-595 | DOI | MR | Zbl

[18]H. Davenport and D. J. Lewis, Two additive equations, Number Theory (Proc. Symp. Pure Math., vol. XII, Houston, Texas, 1967), 74-98, Amer. Math. Soc. Providence, R.I., 1969 | MR | Zbl

[19]J.-M. Deshouillers, Amélioration de la constante de Šnirelman dans le problème de Goldbach, Sém. Delange-Pisot-Poitou 1972/73, Fasc. 2, n° 17, Paris, 1973 | EuDML | Numdam | MR | Zbl

[20]J.-M. Deshouillers, Sur la constante de Šnirel'man, Sém. Delange-Pisot-Poitou 1975/76, Fasc. 2, exp. n° G16, Paris, 1977 | EuDML | Numdam | MR | Zbl

[21]X.-X. Ding, C.-D. Pan and Y. Wang, On the representation of every large even integer as a sum of a prime and an almost prime, Sci. Sinica 28 (1975), 599-610 | MR | Zbl

[22]F. Dress, Amélioration de la majoration de $g (4)$ dans le problème de Waring : $g (4) \leq 34$ , Sém. Delange-Pisot-Poitou 1969/70, fasc. 1, exp. 15, Paris, 1970 | EuDML | MR | Zbl

[23]F. Dress, Sur le problème de Waring pour les puissances quatrièmes, C.R. Acad. Sci. Paris, Sér. A, 272 (1971), 457-459 | MR | Zbl

[24] W. J. Ellison, Waring's problem, Amer. Math. Monthly 78 (1971), 10-36 | DOI | MR | Zbl

[25] 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 | DOI | JFM | MR

[26] S. W. Graham, Applications of sieve methods, Dissertation, University of Michigan, Ann Arbor, 1977 | MR

[27] H. Halberstam, Additive Number Theory, Proc. Springer Lecture Notes (à paraître) | Zbl

[28] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press, London, 1974 | MR | Zbl

[29] H. Halberstam and K. F. Roth, Sequences, vol. 1, Clarendon Press, Oxford, 1966 | Zbl

[30] D. R. Heath-Brown and S. J. Patterson, paper in preparation

[31] A. A. Karatsuba and N. M. Korobov, A mean-value theorem, Dokl. Akad. Nauk SSSR 149 (1963), 245-248 | MR | Zbl

[32] N. I. Klimov, Apropos the computations of Shnirel'man's constant, Volzh.Mat. Sb. Vyp. 7 (1969), 32-40 | MR

[33] N. I. Klimov, G. Z. Pil'Tai and T. A. Sheptitskaya, Estimation of the absolute constant in the Goldbach-Shnirel'man problem, Issled. Teor.Chisel, Saratov4 (1972), 35-51 | Zbl

[34] J. L. Lagrange, Nouv. Mém. Acad. Roy. Soc. Berlin 1770 (1772), 123-133

[35] Ju. V. Linnik, On the representation of large numbers as sums of seven cubes, Mat. Sb. 12 (54) (1943), 218-224 | EuDML | MR | Zbl

[36] Ju. V. Linnik, On the possibility of a unique method in certain problems of "additive" and "distributive" prime number theory, Dokl. Akad. Nauk SSSR 48 (1945), 3-7 | MR | Zbl

[37] Ju. V. Linnik, A new proof of the Goldbach-Vinogradov theorem, Mat.Sb. 19 (61) (1946), 3-8 | EuDML | MR | Zbl

[38] K. Mahler, On the fractional parts of the powers of a rational number II, Mathematika 4 (1957), 122-124 | DOI | MR | Zbl

[39] K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math.Soc. 8 (1968), 313-321 | DOI | MR | Zbl

[40] H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Math. 227 (1971), Berlin and New York | MR | Zbl

[41] H. L. Montgomery and R. C. Vaughan, Error terms in additive prime number theory, Quart. J. Math. Oxford (2) 24 (1973), 207-216 | DOI | MR | Zbl

[42] H. L. Montgomeryand R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arithmetica 27 (1975), 353-370 | DOI | EuDML | MR | Zbl

[43] V. Narasimhamurti, On Waring's problem for 8th, 9th and 10th powers, J. Indian Math. Soc. 5 1941), 122 | MR | Zbl

[44] P. M. Ross, On Chen's theorem that each large even number has the form $p_{1} + p_{2}$ or $p_{1} + p_{2} p_{3}$ , J. London Math. Soc. (2) 10 (1975), 500-506 | DOI | MR | Zbl

[45] P. M. Ross, A short intervals result in additive prime number theory, J. London Math. Soc. (2) 17 (1978), 219-227 | DOI | MR | Zbl

[46] K. Sambasiva Rao, On Waring's problem for smaller powers, J. Indian Math. Soc. 5 (1941), 117-121 | MR | Zbl

[47] L. G. Shnirelman, On additive properties of numbers, Izv. Donskowo Politeh. Inst., 14 (1930), 3-28

[48] L. G. Shnirelman, Über additive Eigenschaften von Zahlen, Math. Ann. 107 (1933), 649-690 | DOI | EuDML | JFM | MR | Zbl

[49] H. Siebert, Darstellung als Summe von Primzahlen (Diplomarbeit, Marburg, 1968)

[50] R. M. Stemmler, The ideal Waring theorem for exponents $401 - 200000$ , Math. Comp. 18 (1964), 144-146 | MR | Zbl

[51] H. E. Thomas Jr., A numerical approach to Waring's problem for fourth powers, Dissertation, University of Michigan, Ann Arbor, 1973 | MR

[52] H. E. Thomas Jr., Waring's problem for twenty-two biquadrates, Trans. Amer. Math. Soc, 193 (1974) | MR | Zbl

[53] K.-C. Tong, On Waring's problem, Advancement in Math. 3 (1957), 602-607 | MR

[54] R. C. Vaughan, On Goldbach's problem, Acta Arithmetica 22 (1972), 21-48 | DOI | EuDML | MR | Zbl

[55] R. C. Vaughan, A new estimate for the exceptional set in Goldbach's problem, Analytic Number Theory (Proc. Symp. Pure Math., vol. XXIV, St. Louis, Missouri, 1972), 315-320, Amer. Math. Soc. Providence, R.I., 1973 | DOI | MR | Zbl

[56] R. C. Vaughan, A note on Shnirel'man's approach to Goldbach's problem, Bull. London Math. Soc., 8 (1976), 245-250 | DOI | MR | Zbl

[57] R. C. Vaughan, On pairs of additive cubic equations, Proc. London Math.Soc. (3) 34 (1977); 354-364 | DOI | MR | Zbl

[58] R. C. Vaughan, On the estimation of Shnirel'man's constant, J. fur Reine Angew. Math., 290 (1977), 93-108 | EuDML | MR | Zbl

[59] R. C. Vaughan, Homogeneous additive equations and Waring's problem, Acta Arithmetica, 33 (1977), 231-253 | DOI | EuDML | MR | Zbl

[60] R. C. Vaughan, Sommes trigonométriques sur les nombres premiers, C.R. Acad. Sci. Paris, Sér. A, 258 (1977), 981-983 | MR | Zbl

[61] R. C. Vaughan, An elementary method in prime number theory, Acta Arithmetica, to appear | EuDML | MR | Zbl

[62] I. M. Vinogradov, Some theorems concerning the theory of primes, Recueil Math. 2 (44), 2 (1937), 179-195 | Zbl

[63] I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Trav. Inst. Math. Steklov 23 (1947) | MR | Zbl

[64] I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, translated from the Russian, revised an annotated by K.F. Roth and A. Davenport, Interscience, London, 1954 | MR | Zbl

[65] I. M. Vinogradov, On an upper bound for $G (n)$ , Izv. Akad. Nauk SSSR 23 (1959), 637-642 | MR | Zbl

[66] G. L. Watson, A proof of the seven cube theorem, J. London Math. Soc. 26 (1951), 153-156 | DOI | MR | Zbl