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 \(\sigma=\operatorname{Re} s=1\).


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


Zbl 0918.11052
Full Text: DOI Euclid


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