×

zbMATH — the first resource for mathematics

On the Goldbach conjecture in arithmetic progressions. (English) Zbl 1148.11053
For integers \(k\), \(b_1\), \(b_2\) and \(b_3\) with \(k\geq1\) and \((b_1b_2b_3,k)=1\), write \(J(N)=J(N;k,b_1,b_2,b_3)\) for the number of representations of \(N\) in the form \(N=p_1+p_2+p_3\) with primes \(p_i\) satisfying \(p_i\equiv b_i\pmod k\) for \(i=1\), 2, 3.
The main theorem of this paper states that for every sufficiently large odd \(N\) satisfying \(N\equiv b_1+b_2+b_3\pmod k\), one has \(J(N)>0\) for all primes \(k\leq R\) with at most \(O((\log N)^B)\) exceptions, where \(R=N^{5/48-\varepsilon}\) with any fixed \(\varepsilon>0\), and \(B\) is a certain positive constant. J. Y. Liu [Chin. Ann. Math. 19, 479–488 (1998; Zbl 0918.11052)] proved a similar result for all but \(O(R(\log N)^{-A})\) primes \(k\leq R\) with \(R=N^{3/20}(\log N)^{-A'}\), where \(A\) and \(A'\) are fixed positive constants. Thus the significance of this work is not primarily on the size of \(R\), but on the amazingly small upper bound for the number of exceptional primes \(k\). The highlight of the paper is skilful treatment of zeros of Dirichlet \(L\)-functions \(L(s,\chi)\) quite near to the line \(\sigma=\operatorname{Re} s=1\).

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11L07 Estimates on exponential sums
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] R. Ayoub, On Rademacher’s extension of the Goldbach-Vinogradov theorem , Canad. J. Math. (1953), 482-491. JSTOR: · Zbl 0050.27001 · doi:10.2307/1990813 · links.jstor.org
[2] C. Bauer, On Goldbach’s conjecture in arithmetic progressions , Stud. Sci. Math. Hungar. 37 (2001), 1-20. · Zbl 0980.11048 · doi:10.1556/SScMath.37.2001.1-2.1
[3] ——–, On the exceptional set for the sum of a prime and the \(k\)-th power of a prime , Stud. Sci. Math. 35 (1999), 291-330. · Zbl 0980.11047
[4] P.X. Gallagher, A large sieve density estimate near \(\sigma=1\) , Invent. Math. 11 (1970), 329-339. · Zbl 0219.10048 · doi:10.1007/BF01403187 · eudml:142061
[5] D.R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan’s identity , Canad. J. Math. 34 (1982), 1365-1377. · Zbl 0478.10024 · doi:10.4153/CJM-1982-095-9
[6] N.M. Huxley, Large values of Dirichlet polynomials III, Acta Arith. 26 (1975), 435-444. · Zbl 0268.10026 · eudml:205330
[7] Jianya Liu, The Goldbach-Vinogradov theorem with three primes in a thin subset , Chinese Annals Math. 19 (1998), 479-488. · Zbl 0918.11052
[8] J.Y. Liu, On Lagrange’s theorem with prime variables , Quart. J. Math. 54 December (2003), 453-462. · Zbl 1080.11071 · doi:10.1093/qjmath/54.4.453
[9] J.Y. Liu and T. Zhan, The ternary Goldbach problem in arithmetic progressions , Acta Arith. 82 (1997), 197-227. · Zbl 0889.11035 · eudml:207088
[10] J.Y. Liu and M.C. Liu, The exceptional set in the four prime squares problem , Illinois J. Math. 44 (2000), 272-293. · Zbl 0942.11044
[11] M.C. Liu and T. Zhan, The Goldbach problem with primes in arithmetic progressions , in Analytic number theory (Yoichi Motohashi, ed.), London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 227-251. · Zbl 0913.11043
[12] H.L. Montgomery and R.C. Vaughan, On the exceptional set in Goldbach‘s problem , Acta Arith. 27 (1975), 353-370. · Zbl 0301.10043 · eudml:205349
[13] Chengdong Pan and Chengbiao Pan, Analytic number theory , Science Press, Beijing, 1992 (in Chinese).
[14] ——–, Goldbach conjecture , Science Press, Beijing, 1991 (in English).
[15] K. Prachar, Primzahlverteilung , Springer Verlag, Berlin, 1978.
[16] H.A. Rademacher, Ueber eine Erweiterung des Goldbachschen problems , Math. Z. 25 (1926). · JFM 52.0167.04
[17] I.M. Vinogradov, Representation of an odd number as the sum of three primes , Dokl. Akad. Nauk. SSSR 15 (1937), 291-294. · Zbl 0016.29101
[18] A. Zaccagnini, The exceptional set for the sum of a prime and a \(k\)-th power , Mathematika 39 (1992), 400-421 · Zbl 0760.11026 · doi:10.1112/S0025579300015114
[19] T. Zhan, On the representation of a large odd integer as a sum of three almost equal primes , Acta Math. Sinica 7 (1991), 259-272. · Zbl 0742.11048 · doi:10.1007/BF02583003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.