## 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

Zbl 0918.11052
Full Text:

### References:

 [1] R. Ayoub, On Rademacher’s extension of the Goldbach-Vinogradov theorem , Canad. J. Math. (1953), 482-491. JSTOR: · Zbl 0050.27001 [2] C. Bauer, On Goldbach’s conjecture in arithmetic progressions , Stud. Sci. Math. Hungar. 37 (2001), 1-20. · Zbl 0980.11048 [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 [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 [6] N.M. Huxley, Large values of Dirichlet polynomials III, Acta Arith. 26 (1975), 435-444. · Zbl 0268.10026 [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 [9] J.Y. Liu and T. Zhan, The ternary Goldbach problem in arithmetic progressions , Acta Arith. 82 (1997), 197-227. · Zbl 0889.11035 [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 [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 [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
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