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The value of additive forms at prime arguments. (English) Zbl 1047.11095
Let \(k\) and \(s\) be integers satisfying either \(k\geq3\) and \(s=2^{k-1}+1\), or \(k\geq6\) and \(s=7\cdot2^{k-4}+1\), let \(\lambda_1,\ldots,\lambda_s\) and \(\delta\) be positive real numbers, and suppose that \(\lambda_1/\lambda_2\) is irrational and algebraic. Write \(E(X)\) for the measure of the set of positive real numbers \(v\leq X\) such that the inequality \[ | \lambda_1p_1+\cdots+\lambda_sp_s-v| <v^{-\delta} \] has no solution in primes \(p_1,\ldots,p_s\). Then the main theorem of this paper asserts that for any \(\varepsilon>0\), one has \(E(X)\ll X^{1-2^{3-2k}(3k)^{-1}+2\delta+\varepsilon}\). This work may be regarded as a generalisation of that of R. J. Cook and A. Fox [Mathematika 48, No. 1–2, 137–149 (2001; Zbl 1035.11010)], in which the above conclusion was shown for \(k=2\). For the case \(k=1\), see J. Brüdern, R. J. Cook and A. Perelli [Sieve methods, exponential sums, and their applications in number theory, Lond. Math. Soc. Lect. Note Ser. 237, 87–100 (1997; Zbl 0924.11085)].
The proof is based on the method of Davenport and Heilbronn, and the minor arc integral is estimated by means of Hua’s lemma, Heath-Brown’s improvement of the latter lemma, and Harman’s estimate for exponential sums over primes.

11P32 Goldbach-type theorems; other additive questions involving primes
11D75 Diophantine inequalities
11P55 Applications of the Hardy-Littlewood method
Full Text: DOI Numdam EuDML EMIS
[1] Bauer, C., Liu, M.-C., Zhan, T., Personal Communication.
[2] Brüdern, J., Cook, R.J., Perelli, A., 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 Press1996, 87-100. · Zbl 0924.11085
[3] Cook, R.J., Fox, A., The values of ternary quadratic forms at prime arguments. Mathematika, to appear. · Zbl 1035.11010
[4] Davenport, H., Indefinite quadratic forms in many variables. Mathematika, 3 (1956), 81-101. · Zbl 0072.27205
[5] Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities. Campus Publishers, Ann Arbor, Michigan, 1962. · Zbl 1089.11500
[6] Davenport, H., Heilbronn, H., On indefinite quadratic forms in five variables. J. London Math. Soc.21 (1946), 185-193. · Zbl 0060.11914
[7] Davenport, H., Roth, K.F., The solubility of certain diophantine inequalities. Mathematika, 2 (1955), 81-96. · Zbl 0066.29301
[8] Ghosh, A., The distribution of αp2 modulo 1. Proc. London Math. Soc. (3) 42 (1981), 252-269. · Zbl 0447.10035
[9] Harman, G., Trigonometric sums over primes. Mathematika28 (1981), 249-254. · Zbl 0465.10029
[10] Hardy, G.H., Littlewood, J.E., Some problems of “Partitio Numerorum” , V. Proc. London Math. Soc. (2) 22 (1923), 46-56. · JFM 49.0127.03
[11] Heath-Brown, D.R., Weyl’s inequality, Hua’s inequality and Waring’s problem. J. London Math. Soc38 (1988), 216-230. · Zbl 0619.10046
[12] Hua, L.K., Some results in the additive prime number theory. Quart. J. Math. Oxford9 (1938), 68-80. · JFM 64.0131.02
[13] Hua, L.K., On Waring’s problem. Quart. J. Math. Oxford9 (1938), 199-202. · JFM 64.0124.04
[14] Leung, M.-C., Liu, M.-C., On generalized quadratic equations in three prime variables. Monatsh. Math.115 (1993), 113-169. · Zbl 0779.11045
[15] Li, H., The exceptional set of Goldbach numbers. Quart. J. Math Oxford50 (1999), 471-482. · Zbl 0937.11046
[16] Li, H., The exceptional set of Goldbach numbers II. Preprint. · Zbl 0963.11057
[17] Montgomery, H.L., Vaughan, R.C., The exceptional set in Goldbach’s problem. Acta Arith.27 (1975), 353-370. · Zbl 0301.10043
[18] Schwarz, W., Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen, II. J. Reine Angew. Math.206 (1961), 78-112. · Zbl 0102.28201
[19] Vaughan, R.C., Diophantine approximation by prime numbers I. Proc. London Math. Soc. (3) 28 (1974), 373-384. · Zbl 0274.10045
[20] Vaughan, R.C., Diophantine approximation by prime numbers II. Proc. London Math. Soc. (3) 28 (1974), 385-401. · Zbl 0276.10031
[21] Watson, G.L., On indefinite quadratic forms in five variables. Proc. London Math. Soc. (3) 3 (1953), 170-181. · Zbl 0050.04704
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