On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 925-948.

Nous considérons une version affaiblie de la conjecture sur la représentation des entiers comme somme de quatre carrés de nombres premiers.

We consider an approximation to the popular conjecture about representations of integers as sums of four squares of prime numbers.

@article{JTNB_2005__17_3_925_0,
author = {Tolev, Doychin},
title = {On the exceptional set of {Lagrange{\textquoteright}s} equation with three prime and one almost{\textendash}prime variables},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {925--948},
publisher = {Universit\'e Bordeaux 1},
volume = {17},
number = {3},
year = {2005},
doi = {10.5802/jtnb.528},
mrnumber = {2212133},
zbl = {05016595},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.528/}
}
TY  - JOUR
AU  - Tolev, Doychin
TI  - On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 925
EP  - 948
VL  - 17
IS  - 3
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.528/
UR  - https://www.ams.org/mathscinet-getitem?mr=2212133
UR  - https://zbmath.org/?q=an%3A05016595
UR  - https://doi.org/10.5802/jtnb.528
DO  - 10.5802/jtnb.528
LA  - en
ID  - JTNB_2005__17_3_925_0
ER  - 
Tolev, Doychin. On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 925-948. doi : 10.5802/jtnb.528. http://www.numdam.org/articles/10.5802/jtnb.528/

[1] C. Bauer, M.-C. Liu, T. Zhan, On a sum of three prime squares. J. Number Theory 85 (2000), 336–359. | MR 1802721 | Zbl 0961.11034

[2] J. Brüdern, E. Fouvry, Lagrange’s Four Squares Theorem with almost prime variables. J. Reine Angew. Math. 454 (1994), 59–96. | MR 1288679 | Zbl 0809.11060

[3] G. Greaves, On the representation of a number in the form ${x}^{2}+{y}^{2}+{p}^{2}+{q}^{2}$ where $p$ and $q$ are odd primes. Acta Arith. 29 (1976), 257–274. | MR 404182 | Zbl 0283.10030

[4] H. Halberstam, H.-E. Richert, Sieve methods. Academic Press, 1974. | MR 424730 | Zbl 0298.10026

[5] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers. Fifth ed., Oxford Univ. Press, 1979. | MR 568909 | Zbl 0423.10001

[6] G. Harman, A. V. Kumchev, On sums of squares of primes. Math. Proc. Cambridge Philos. Soc., to appear. | MR 2197572 | Zbl 05012459

[7] D.R. Heath-Brown, Cubic forms in ten variables. Proc. London Math. Soc. 47 (1983), 225–257. | MR 703978 | Zbl 0494.10012

[8] D.R. Heath-Brown, D.I.Tolev, Lagrange’s four squares theorem with one prime and three almost–prime variables. J. Reine Angew. Math. 558 (2003), 159–224. | MR 1979185 | Zbl 1022.11050

[9] L.K. Hua, Some results in the additive prime number theory. Quart. J. Math. Oxford 9 (1938), 68–80. | Zbl 0018.29404

[10] L.K. Hua, Introduction to number theory. Springer, 1982. | MR 665428 | Zbl 0483.10001

[11] L.K. Hua, Additive theory of prime numbers. American Mathematical Society, Providence, 1965. | MR 194404 | Zbl 0192.39304

[12] H. Iwaniec, Rosser’s sieve. Acta Arith. 36 (1980), 171–202. | MR 581917 | Zbl 0435.10029

[13] H. Iwaniec, A new form of the error term in the linear sieve. Acta Arith. 37 (1980), 307–320. | MR 598883 | Zbl 0444.10038

[14] H.D. Kloosterman, On the representation of numbers in the form $a{x}^{2}+b{y}^{2}+c{z}^{2}+d{t}^{2}$. Acta Math. 49 (1926), 407–464.

[15] J. Liu, On Lagrange’s theorem with prime variables. Quart. J. Math. Oxford, 54 (2003), 453–462. | MR 2031178 | Zbl 1080.11071

[16] J. Liu, M.-C. Liu, The exceptional set in the four prime squares problem. Illinois J. Math. 44 (2000), 272–293. | MR 1775322 | Zbl 0942.11044

[17] J.Liu, T. D. Wooley, G. Yu, The quadratic Waring–Goldbach problem. J. Number Theory, 107 (2004), 298–321. | MR 2072391 | Zbl 1056.11055

[18] V.A. Plaksin, An asymptotic formula for the number of solutions of a nonlinear equation for prime numbers. Math. USSR Izv. 18 (1982), 275–348. | Zbl 0482.10045

[19] P. Shields, Some applications of the sieve methods in number theory. Thesis, University of Wales 1979.

[20] D.I. Tolev, Additive problems with prime numbers of special type. Acta Arith. 96, (2000), 53–88. | MR 1812750 | Zbl 0972.11096

[21] D.I. Tolev, Lagrange’s four squares theorem with variables of special type. Proceedings of the Session in analytic number theory and Diophantine equations, Bonner Math. Schriften, Bonn, 360 (2003). | MR 2075638 | Zbl 1060.11061

[22] T.D. Wooley, Slim exceptional sets for sums of four squares, Proc. London Math. Soc. (3), 85 (2002), 1–21. | MR 1901366 | Zbl 1039.11066

Cité par Sources :