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: \[ A\{\xi: \vert \xi\vert < x^{1/n}, \vert v(\xi) - \log\log x\vert> (\log\log x)^{1/n} f(x)\} = O\left(\frac{x}{f^2(x)}\right). \]