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Übertragung des Goldbach-Vinogradovschen Satzes auf reell-quadratische Zahlkörper. (German) Zbl 0099.03602
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.
Reviewer: M. Cugiani

11P32 Goldbach-type theorems; other additive questions involving primes
11R11 Quadratic extensions
Full Text: DOI EuDML
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