A consequence of the ternary Goldbach theorem (Q2803020)

Let \(\mathbb{N} ,\mathbb{C} \) and \( P\) be the set of positive integers, complex and prime numbers, respectively. For each \(k \in \mathbb{N}\), let \( M_{k}=\{p_{1}+p_{2}+\ldots+p_{k}:p_{1},p_{2},\ldots,p_{k} \in P \} \). The Goldbach conjecture states that every even integer \(n \geq 4\) can be written as the sum of two primes, and the ternary Goldbach conjecture is that every odd integer larger than 5 is the sum of three primes. In 1923, \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Proc. Lond. Math. Soc. (2) 22, 46--56 (1923; JFM 49.0127.03)] ] used the newly formulated circle method discovered by Hardy and Ramanujan to show that the ternary Goldbach problem has a solution if we assume the generalized Riemann hypothesis for Dirichlet's \(L\)-functions. In 1937, \textit{I. M. Vinogradov} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 15, 169--172 (1937; Zbl 0016.29101)] introduced a new method of trigonometric sums to show that the ternary Goldbach conjecture is true in weak form, namely that every large odd number is a sum of three primes. His result gave no lower bound for what ``sufficiently large'' meant. In 2002, \textit{M.-C. Liu} and \textit{T. Wang} [Acta Arith. 105, No. 2, 133--175 (2002; Zbl 1019.11026)] gave the bound \(\exp(3100)\) and finally \textit{H. A. Helfgott} [Gac. R. Soc. Mat. Esp. 16, No. 4, 709--726 (2013; Zbl 1296.11129)] proved recently that the ternary Goldbach conjecture is true, i.e. every odd integer \(n \geq 7\) belongs to \( M_{3}\). He proved somewhat more that if \(n \geq 9\) and \(n\) is odd then \(n\) is a sum of three odd primes.NEWLINENEWLINENEWLINEIn this paper, it is proved that if \(k \geq 4\) is an integer and \(f:M_{k}\to \mathbb{C}\), \(g: P\to \mathbb{C}\) are functions such that NEWLINE\[NEWLINEf(p_{1}+p_{2}+...+p_{k})=g(p_{1})+g(p_{2})+...+g(p_{k})NEWLINE\]NEWLINE holds for every \(p_{1},\ldots,p_{k} \in P\), then there exist suitable constants \(A,B \in \mathbb{C}\) such that NEWLINE\[NEWLINEf(n)=An+kB\quad\text{for}\;n\in \mathbb{N},\;n \geq 2kNEWLINE\]NEWLINE and NEWLINE\[NEWLINE g(p)=Ap+B \quad\text{for}\;p\in P.NEWLINE\]NEWLINE In the end some conjectures have been stated.