Über die Anzahl der Primfaktoren algebraischer Zahlen und das Gaußsche Fehlergesetz. (German) Zbl 0286.12002
Let \(x\) be an arbitrary positive number, \(K\) an algebraic number field of degree \(n\), \(\mathfrak f\) an integral ideal in \(K\) of norm \(N\mathfrak f\), \(A\{\mathfrak p: \ldots\}\) the number of prime ideals \(\mathfrak p\) of \(K\) with \(\ldots\), \(\xi\) an integer in \(K\) of norm \(N\xi\), \(\xi^j\) \((j = 1,2, \ldots, n)\) the conjugates of \(\xi\), \(A\{\xi: \ldots\}\) the number of the integers \(\xi\) of \(K\) with \(\ldots\); \(\vert \xi\vert < x^{1/n}\) denotes the system \(\vert \xi^j\vert \le x^{1/n}\) \((j = 1,2, \ldots, n)\), \(c_1, c_2, \ldots\) are positive constants dependent on \(K\). Further, let \(\displaystyle{\sum_\xi}'\) mean \(\displaystyle\sum_{\xi\ne 0}\); \(v(\xi) = A\{\mathfrak p: \mathfrak p\mid \xi\}\). The author proves a number of theorems, some of which are mentioned below.
Theorem 1: \(x, \alpha, \beta\) are arbitrary real numbers with \(x > c_3\) and \(\alpha<\beta\). Let \(\mathfrak f\) be an integral ideal of \(K\) with \(N\mathfrak f < e^{(\log \log x)^2}\), \(\gamma\) an integer of \(K\), and \(\mu = \max(x, \vert\alpha\vert, \vert\beta\vert)\). Then for \(z = x\) and \(z= \vert N\xi\vert\) \((z > e)\), one has the formula \[ \begin{gathered} A\{\xi: \xi \equiv\gamma\bmod \mathfrak f, \vert \xi\vert < x^{1/n}, \log \log z + \alpha(\log \log z)^{1/2} < v(\xi)< \log \log z + \beta(\log \log z)^{1/2}= \\ \frac{c_1x}{(2\pi)^{1/2}\cdot N\mathfrak f} \int_\alpha^\beta e^{-u^2/2}\,du + O\left(\frac{\mu^4 x(\log \log \log x)^{1/2}} {N\mathfrak f \cdot (\log \log x)^{1/4}}\right). \end{gathered} \]
Theorem 3 and Theorem 5: For each real number \(x\ge 3\), each integral ideal \(\mathfrak f\) of \(K\) with \(N\mathfrak f\le x\) and each integer \(\gamma\) of \(K\), \[ {\sum_{\substack{\vert \xi\vert < x^{1/n} \\ \equiv \gamma\bmod\mathfrak f}}}' v(\xi) = \\ \frac{c_1x}{ N\mathfrak f} \log \log x + O\left(\frac{x}{N\mathfrak f}\log N\mathfrak f\right) + O\left(\frac{x}{\log x}N\mathfrak f^{1/n-1}\right), \] Finally, the function \(f(x)>0\) tends monotonically to infinity. \[ \begin{split} {\sum_{\substack{\vert \xi\vert < x^{1/n} \\ \equiv \gamma\mod\mathfrak f}}}' v^2(\xi) = \frac{c_1x}{ N\mathfrak f} (\log \log x)^2 + O\left(\frac{x}{N\mathfrak f} \log N\mathfrak f\cdot \log \log x \right) + \\ O\left(\frac{x}{N\mathfrak f} \log^2 N\mathfrak f\right) + O\left(xN\mathfrak f^{1/n - 1} \frac{\log\log x}{\log x}\right). \end{split} \]
Theorem 6: \[ A\{\xi: \vert \xi\vert < x^{1/n}, \vert v(\xi) - \log\log x\vert> (\log\log x)^{1/n} f(x)\} = O\left(\frac{x}{f^2(x)}\right). \]
Theorem 1: \(x, \alpha, \beta\) are arbitrary real numbers with \(x > c_3\) and \(\alpha<\beta\). Let \(\mathfrak f\) be an integral ideal of \(K\) with \(N\mathfrak f < e^{(\log \log x)^2}\), \(\gamma\) an integer of \(K\), and \(\mu = \max(x, \vert\alpha\vert, \vert\beta\vert)\). Then for \(z = x\) and \(z= \vert N\xi\vert\) \((z > e)\), one has the formula \[ \begin{gathered} A\{\xi: \xi \equiv\gamma\bmod \mathfrak f, \vert \xi\vert < x^{1/n}, \log \log z + \alpha(\log \log z)^{1/2} < v(\xi)< \log \log z + \beta(\log \log z)^{1/2}= \\ \frac{c_1x}{(2\pi)^{1/2}\cdot N\mathfrak f} \int_\alpha^\beta e^{-u^2/2}\,du + O\left(\frac{\mu^4 x(\log \log \log x)^{1/2}} {N\mathfrak f \cdot (\log \log x)^{1/4}}\right). \end{gathered} \]
Theorem 3 and Theorem 5: For each real number \(x\ge 3\), each integral ideal \(\mathfrak f\) of \(K\) with \(N\mathfrak f\le x\) and each integer \(\gamma\) of \(K\), \[ {\sum_{\substack{\vert \xi\vert < x^{1/n} \\ \equiv \gamma\bmod\mathfrak f}}}' v(\xi) = \\ \frac{c_1x}{ N\mathfrak f} \log \log x + O\left(\frac{x}{N\mathfrak f}\log N\mathfrak f\right) + O\left(\frac{x}{\log x}N\mathfrak f^{1/n-1}\right), \] Finally, the function \(f(x)>0\) tends monotonically to infinity. \[ \begin{split} {\sum_{\substack{\vert \xi\vert < x^{1/n} \\ \equiv \gamma\mod\mathfrak f}}}' v^2(\xi) = \frac{c_1x}{ N\mathfrak f} (\log \log x)^2 + O\left(\frac{x}{N\mathfrak f} \log N\mathfrak f\cdot \log \log x \right) + \\ O\left(\frac{x}{N\mathfrak f} \log^2 N\mathfrak f\right) + O\left(xN\mathfrak f^{1/n - 1} \frac{\log\log x}{\log x}\right). \end{split} \]
Theorem 6: \[ A\{\xi: \vert \xi\vert < x^{1/n}, \vert v(\xi) - \log\log x\vert> (\log\log x)^{1/n} f(x)\} = O\left(\frac{x}{f^2(x)}\right). \]
Reviewer: B. K. Ghosh
MSC:
11R47 | Other analytic theory |
11N05 | Distribution of primes |
11K99 | Probabilistic theory: distribution modulo \(1\); metric theory of algorithms |
Keywords:
generalization of Ramanujan-Erdös-Kac theorem; average number of prime factors; algebraic number fieldReferences:
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[4] | Tanaka, Jap. J. Math. 25 pp 1– (1955) |
[5] | J. Math. Soc. Japan 9 pp 171– (1957) |
[6] | Jap. J. Math. 27 pp 103– (1957) |
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