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On Linnik’s approximation to Goldbach’s problem. II. (English) Zbl 1474.11173

Yu. V. Linnik [Tr. Mat. Inst. Steklova 38, 152–169 (1951; Zbl 0049.31402)] proved the existence of a constant \(K\) such that every sufficiently large even number is the sum of two primes and at most \(K\) powers of 2. D. R. Heath-Brown and J. C. Puchta [Asian J. Math. 6, No. 3, 535–566 (2002; Zbl 1097.11050)] found that \(K = 13\) works. This was improved to \(K=8\) by J. Pintz and I. Z. Ruzsa [Acta Arith. 109, No. 2, 169–194 (2003; Zbl 1031.11060)]. In the present work they show that \(K = 8\) powers of 2 suffices without supposing any unproved hypothesis.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11L07 Estimates on exponential sums
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References:

[1] Chen, JR, On Goldbach’s problem and the sieve methods, Sci. Sinica Ser. A, 21, 701-738 (1978) · Zbl 0399.10046
[2] Davenport, H., Multiplicative Number Theory, Markham Publishing Co (1967), Chicago: Ill, Chicago · Zbl 0159.06303
[3] Gallagher, PX, Primes and powers of 2, Invent. Math., 29, 125-142 (1975) · Zbl 0305.10044 · doi:10.1007/BF01390190
[4] Goldfeld, DM; Schinzel, A., On Siegel’s zero, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2, 571-583 (1975) · Zbl 0327.10041
[5] Heath-Brown, DR; Puchta, J-C, Integers represented as a sum of primes and powers of two, Asian J. Math., 6, 535-565 (2002) · Zbl 1097.11050 · doi:10.4310/AJM.2002.v6.n3.a7
[6] Iwaniec, H., On zeros of Dirichlet’s L-series, Invent. Math., 23, 97-104 (1974) · Zbl 0275.10024 · doi:10.1007/BF01405163
[7] Jutila, M., On Linnik’s constant, Math. Scand., 41, 45-62 (1975) · Zbl 0363.10026 · doi:10.7146/math.scand.a-11701
[8] A. Khalfalah and J. Pintz, On the representation of Goldbach numbers by a bounded number of powers of two, in: Elementare und analytische Zahlentheorie, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, vol. 20 (2006), pp. 129-142 · Zbl 1177.11085
[9] Li, HZ, The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes, Acta Arith., 92, 229-237 (2000) · Zbl 0952.11022 · doi:10.4064/aa-92-3-229-237
[10] H. Z. Li, The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes. (II), Acta Arith., 96 (2001), 369-379 · Zbl 0973.11088
[11] Linnik, YuV, Prime numbers and powers of two, Trudy Mat. Inst. Steklov, 38, 152-169 (1951) · Zbl 0049.31402
[12] Yu. V. Linnik, Addition of prime numbers with powers of one and the same number, Mat. Sb. (N.S.), 32 (1953), 3-60 (in Russian) · Zbl 0051.03402
[13] Liu, Z.; Liu, G., Density of two squares of primes and powers of two, Int. J. Number Theory, 7, 1317-1329 (2011) · Zbl 1237.11042 · doi:10.1142/S1793042111004605
[14] J. Y. Liu, M. C. Liu and T. Z. Wang, The number of powers of \(2\) in a representation of large even integers. (I), Sci. China Ser. A, 41 (1998), 386-397 · Zbl 1029.11049
[15] J. Y. Liu, M. C. Liu and T. Z. Wang, The number of powers of \(2\) in a representation of large even integers. (II), Sci. China Ser. A, 41 (1998), 1255-1271 · Zbl 0924.11086
[16] Liu, JY; Liu, MC; Wang, TZ, On the almost Goldbach problem of Linnik, J. Théor. Nombres Bordeaux, 11, 133-147 (1999) · Zbl 0979.11051 · doi:10.5802/jtnb.242
[17] J . Pintz, Elementary methods in the theory of L-functions. II. On the greatest real zero of a real \(L\)-function, Acta Arith., 31 (1976), 273-289 · Zbl 0307.10041
[18] J. Pintz, A new explicit formula in the additive theory of primes with applications, I. The explicit formula for the Goldbach and Generalized Twin Prime problems, arXiv: 1804.05561
[19] J. Pintz, A new explicit formula in the additive theory of primes with applications. II. The exceptional set for the Goldbach problems, arXiv: 1804.09084
[20] Pintz, J.; Ruzsa, IZ, On Linnik’s approximation to Goldbach’s problem, I, Acta Arith., 109, 169-194 (2003) · Zbl 1031.11060 · doi:10.4064/aa109-2-6
[21] Vaughan, RC, On Goldbach’s problem, Acta Arith., 22, 21-48 (1972) · Zbl 0216.31603 · doi:10.4064/aa-22-1-21-48
[22] Vinogradov, IM, Representation of an odd number as a sum of three prime numbers, Doklady Akad. Nauk SSSR, 15, 291-294 (1937) · JFM 63.0131.04
[23] Wang, TZ, On Linnik’s almost Goldbach theorem, Sci. China Ser. A, 42, 1155-1172 (1999) · Zbl 0978.11054 · doi:10.1007/BF02875983
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