Goldbach's conjecture in arithmetic progressions: number and size of exceptional prime moduli
The paper under review investigates the exceptional set \(S_{k,b_1,b_2}(X)\) defined by \[ \{n\leq X: 2\mid n,\ n\equiv b_1+b_2\bmod{k},\ n\ne p_1+p_2 \text{ for any primes } p_i\equiv b_i \bmod{k_i},\ i=1,2\} \] in the binary Goldbach conjecture with primes restricted to arithmetic progressions. The goal of these investigations is to show that this set is small, i.e., of size \[ \sharp S_{k,b_1,b_2}= o\left(X/\varphi(k)\right). \] In [\textit{M.-C. Liu} and \textit{T. Zhan}, Lond. Math. Soc. Lect. Note Ser. 247, 227--251 (1997; Zbl 0913.11043)], estimates of this kind were shown for moduli \(k\) which are a very small power of \(X\). For larger values of \(k\), such estimates were established for almost all moduli \(k\) in [the author, Far East J. Math. Sci. (FJMS) 70, No. 1, 87--120 (2012; Zbl 1348.11076)] and for even larger values of \(k\) they were shown to hold for almost all \(k\) and \(b_1\) and any fixed \(b_2\) in [the author and \textit{Y. Wang}, Acta Arith. 159, No. 3, 227--243 (2013; Zbl 1348.11077)]. The paper under review improves upon these almost-all results by increasing the admissible \(k\)-ranges substantially. The main theorems are as follows. Theorem 1. Set \(R:=X^{1/3-\varepsilon}\). For any constants \(D,U,E>0\), all but \(O( R(\log X)^{-E})\) prime numbers \(k\leq R\), any fixed \(b_2\bmod{k}\), \((k,b_2)=1\), and all but \(O(k(\log k)^{-U})\) integers \(b_1 \bmod{k}\), \((b_1,k)=1\), \[ \sharp S_{k,b_1,b_2}=O(X(\log X)^{-D}k^{-1}). \tag{1} \] Theorem 2. There exists a \(D>0\) such that for all but \(O((\log X)^{D})\) prime numbers \(k\leq X^{5/24-\varepsilon}\), and all integers \(b_1,b_2\) with \((k,b_1b_2)=1\), (1) is true. Theorem 1 improves the exponent \(1/4-\varepsilon\) obtained in the the above-mentioned paper [loc. cit.] to \(1/3-\varepsilon\), and Theorem 2 improves the exponent \(5/48-\varepsilon\) obtained in the above-mentioned paper [loc. cit.] to \(5/24-\varepsilon\). As usual in this context, the circle method is the basic approach. The novelty in the proof of Theorem 1 is a new estimate for exponential sums over primes in arithmetic progressions, valid for almost all primes \(k\leq N^{1/3-\varepsilon}\), which is used in the treatment of the minor arcs. The improvement in Theorem 2 comes from a precise calculation of the contribution of the zeros of Dirichlet \(L\)-functions close to the line \(\mathrm{Re}(s)=1\).
scientific article; zbMATH DE number 6695529