# zbMATH — the first resource for mathematics

Let $$K$$ be a real quadratic field and $$D$$ its discriminant. Denote by $$H$$ the class number of $$K$$ in strict sense [two ideals $$A,B$$ can be considered equivalent if $$A/B=(\kappa)$$, where $$\kappa$$ is totally positive and by $$\varepsilon$$ $$(>1)$$ the generator of the group of totally positive units of $$K$$. Let $$\lambda$$ be a totally positive integer of $$K$$, and denote by $$A_r(\lambda)$$ the number of representations of $$\lambda$$ as sum of $$r$$ totally positive primes of $$K$$. The author proves the following formula: $A_r(\lambda)=\left(\sqrt D/[(r-1)!]^2 (H\log \varepsilon)^r\right)\left((N\lambda)^{r-1}/(\log N\lambda)^r\right) \mathfrak S_r(\lambda)+O\left((N\lambda)^{r-1}/(\log N\lambda)^{r+1}\right).$ [$$N\alpha$$ denotes the norm of $$\alpha\in K$$.] Let $$\mathfrak L$$ be the product of all prime ideals $$\mathfrak p\in K$$ such that $$N\mathfrak p=2$$; in the case that such ideals do not exist in $$K$$ assume $$\mathfrak L=1$$. An integer $$\alpha\in K$$ is called an “even number” if $$\mathfrak L\mid \alpha$$, an “odd number” if $$(\mathfrak L,\alpha)=1$$. It is a result of H. Rademacher [Math. Z. 27, 321–426 (1927; JFM 53.0154.03)] that the “singular series” $$\mathfrak S_r(\lambda)$$ is positive if $$r$$ and $$\lambda$$ are odd and $$r\geq 3$$. In the particular case $$r=3$$, $$\lambda$$ odd, we can deduce the analog of the Goldbach-Vinogradov theorem: for every real quadratic field there exists a constant $$A_0>0$$ such that every totally positive odd integer $$\lambda\in K$$, with $$N\lambda>A_0$$, can be represented as sum of three positive prime integers of $$K$$. In the proof the Vinogradov method of trigonometric sums and a Siegel method, analogous to the Farey dissection, are used.