The value of additive forms at prime arguments

Roger J. Cook

Roger J. Cook

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 77-91.

Abstract (VO)
Résumé

Let $f (𝐩)$ be an additive form of degree $k$ with $s$ prime variables $p_{1}, p_{2}, \dots, p_{s}$ . Suppose that $f$ has real coefficients $λ_{i}$ with at least one ratio $λ_{i} / λ_{j}$ algebraic and irrational. If s is large enough then $f$ takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for $k \geq 6$ , Heath-Brown’s improvement on Hua’s Lemma.

Soit $f (𝐩)$ une forme de degré $k$ en $s$ variables $p_{1}, p_{2}, \dots, p_{s}$ , où les $p_{i}$ sont des nombres premiers. On suppose que les coefficients $λ_{i}$ de $f$ sont réels, et qu’au moins l’un des rapports $λ_{i} / λ_{j}$ est irrationnel et algébrique. Si $s$ est assez grand alors $f$ prend des valeurs proches de tous les termes d’une suite arbitraire de nombres bien-espacés. Cela généralise des travaux antérieurs de Brüdern, Cook et Perelli (formes linéaires), et Cook et Fox (formes quadratiques). La preuve de ce résultat dépend du Lemme de Hua et, pour $k \geq 6$ des améliorations dues à Heath-Brown de ce lemme.

MR Zbl

@article{JTNB_2001__13_1_77_0,
     author = {Roger J. Cook},
     title = {The value of additive forms at prime arguments},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {77--91},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     zbl = {1047.11095},
     mrnumber = {1838071},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_77_0/}
}

TY  - JOUR
AU  - Roger J. Cook
TI  - The value of additive forms at prime arguments
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2001
SP  - 77
EP  - 91
VL  - 13
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_77_0/
LA  - en
ID  - JTNB_2001__13_1_77_0
ER  -

%0 Journal Article
%A Roger J. Cook
%T The value of additive forms at prime arguments
%J Journal de théorie des nombres de Bordeaux
%D 2001
%P 77-91
%V 13
%N 1
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_77_0/
%G en
%F JTNB_2001__13_1_77_0

Roger J. Cook. The value of additive forms at prime arguments. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 77-91. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_77_0/

Bibliographie
Cité par

[1] C. Bauer, M.-C. Liu, T. Zhan, Personal Communication.

[2] J. Brüdern, R.J. Cook, A. Perelli, The values of binary linear forms at prime arguments. Sieve Methods. In Exponential Sums and their Applications in Number Theory, ed. G.R.H. Greaves, G. Harman and M.N. Huxley, Cambridge University Press 1996, 87-100. | MR | Zbl

[3] R.J. Cook, A. Fox, The values of ternary quadratic forms at prime arguments. Mathematika, to appear. | MR | Zbl

[4] H. Davenport, Indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81-101. | MR | Zbl

[5] H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities. Campus Publishers, Ann Arbor, Michigan, 1962. | MR | Zbl

[6] H. Davenport, H. Heilbronn, On indefinite quadratic forms in five variables. J. London Math. Soc. 21 (1946), 185-193. | MR | Zbl

[7] H. Davenport, K.F. Roth, The solubility of certain diophantine inequalities. Mathematika, 2 (1955), 81-96. | MR | Zbl

[8] A. Ghosh, The distribution of αp2 modulo 1. Proc. London Math. Soc. (3) 42 (1981), 252-269. | Zbl

[9] G. Harman, Trigonometric sums over primes. Mathematika 28 (1981), 249-254. | MR | Zbl

[10] G.H. Hardy, J.E. Littlewood, Some problems of "Partitio Numerorum" , V. Proc. London Math. Soc. (2) 22 (1923), 46-56. | JFM

[11] D.R. Heath-Brown, Weyl's inequality, Hua's inequality and Waring's problem. J. London Math. Soc 38 (1988), 216-230. | Zbl

[12] L.K. Hua, Some results in the additive prime number theory. Quart. J. Math. Oxford 9 (1938), 68-80. | JFM | Zbl

[13] L.K. Hua, On Waring's problem. Quart. J. Math. Oxford 9 (1938), 199-202. | JFM | Zbl

[14] M.-C. Leung, M.-C. Liu, On generalized quadratic equations in three prime variables. Monatsh. Math. 115 (1993), 113-169. | MR | Zbl

[15] H. Li, The exceptional set of Goldbach numbers. Quart. J. Math Oxford 50 (1999), 471-482. | MR | Zbl

[16] H. Li, The exceptional set of Goldbach numbers II. Preprint. | MR

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

[18] W. Schwarz, Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen, II. J. Reine Angew. Math. 206 (1961), 78-112. | MR | Zbl

[19] R.C. Vaughan, Diophantine approximation by prime numbers I. Proc. London Math. Soc. (3) 28 (1974), 373-384. | MR | Zbl

[20] R.C. Vaughan, Diophantine approximation by prime numbers II. Proc. London Math. Soc. (3) 28 (1974), 385-401. | MR | Zbl

[21] G.L. Watson, On indefinite quadratic forms in five variables. Proc. London Math. Soc. (3) 3 (1953), 170-181. | MR | Zbl