Some applications of sieve methods in algebraic number fields
Let \(K\) be an algebraic number field of degree \(n=r_ 1+2r_ 2\), with discriminant \(d\) and with ring of integers \(Z_ K\). Denote by \({\mathfrak R}\) the set of integers \(\alpha \in Z_ K\) satisfying \(\alpha\) \(\equiv \gamma \bmod {\mathfrak q}\) (integral ideal), \(Q_ k<\alpha \leq Q_ k+P_ k\) for \(k=1,\ldots,r_ 1\), \(| \alpha^{(k)}| \leq P_ k\) for \(k=r_ 1+1,\ldots,n\). Put \(P=P_ 1\cdot P_ 2\cdots P_ n.\) The main result is an upper estimate for the number of primes \(\omega\in {\mathfrak R}\), for which \(F(\omega)\) is prime, too; here \(F(x)\in Z_ K[X]\) is an irreducible polynomial of degree \(g\geq 1\) with \(F(0)\neq 0\). Denoting by \(\rho(\mathfrak p)\) the number of solutions of \(F(\alpha)\equiv 0 \bmod {\mathfrak p}\) and assuming that \(\rho(\mathfrak p)<N\mathfrak p\) and \(\rho(\mathfrak p)<N\mathfrak p-1\) if \(\mathfrak p\nmid F(0)\) and \(N\mathfrak p\leq g+1\), the author proves for \(1\leq N\mathfrak q<P\) and \((\gamma,\mathfrak p)=1\) the remarkable estimate \[ \sum_{\omega\in {\mathfrak R},\omega,F(\omega)'}1 \leq 8\cdot \frac{| \sqrt{d}|}{(2\pi)^{r_ 2}} \cdot \left(\frac{w}{2^{r_ 1}\cdot hR}\right)^ 2 \cdot \prod_{\mathfrak p\mid\mathfrak q} \left(1-\frac{1}{N{\mathfrak p}}\right)^{-1}\cdot \] \[ \prod_{{\mathfrak p}\nmid {\mathfrak q}}\left(1- \frac{\rho'({\mathfrak p})}{N{\mathfrak p}}\right) \left(1-\frac{1}{N{\mathfrak p}}\right)^{-2} \cdot \frac{P}{\phi(q) \cdot \log^ 2(P/N{\mathfrak q})} \cdot \left\{1+O\left(\frac{\log\log 3P}{\log P/N{\mathfrak q}}\right)\right\}, \] with an \(O\)-constant independent of \(Q_ 1,\ldots,Q_{r_ 1}\) and \(\gamma\), and where \(\rho'({\mathfrak p}) = \rho ({\mathfrak p})\) if \({\mathfrak p}| F(0)\), otherwise \(\rho({\mathfrak p})+1\). Then a good upper estimate for the number of \(\omega\in {\mathfrak R}\) for which \(\omega\) and \(\xi-\omega\) are prime (\(\xi\) is an even algebraic number -- Goldbach problem in algebraic number fields) is deduced. Finally the author gives an asymptotic formula for the number of prime ideals \({\mathfrak p}\) with \(N{\mathfrak p}\leq X\) for which \(\alpha\in {\mathfrak R}\) is a primitive root modulo \({\mathfrak p}\) (disregarding ''few'' exceptional \(\alpha)\).
Goldbach problem in algebraic number fields
scientific article; zbMATH DE number 3882536