\(\Omega \)-estimates related to irreducible algebraic integers (Q2786306)

Let \(K\) be an algebraic number field and \(O_K\) its ring of integers. Denote by \(M_K\) the set of principal ideals of \(O_K\) whose generators are irreducible, and put \(M_K(x)=\#\{I\in M_K:\;N(I)\leq x\}\) and NEWLINE\[NEWLINE\zeta(s,M_K)=\sum_{I\in M_K}{1\over N(I)^s}\quad (\text{Re}\, s>1).NEWLINE\]NEWLINE It was shown by \textit{P. Rémond} [Ann. Sci. Éc. Norm. Sér., III. Sér. 83, 343--410 (1966; Zbl 0157.09602)] that NEWLINE\[NEWLINEM_K(x)=(c(K)+o(1)){x(\log\log x)^{D_K-1}\over\log x},NEWLINE\]NEWLINE where \(c(K)>0\) and \(D_K\) is the Davenport constant of the ideal class-group of \(K\).NEWLINENEWLINEIn 1983 the author obtained an asymptotic expansion of \(M_K(x)\) [Acta Arith., 43, 53--68 (1983; Zbl 0526.12006)], and later [Monatsh. Math., 156, 47--71 (2009; Zbl 1197.11146)] he showed that the same expansion has the integral NEWLINE\[NEWLINET_K(x)={1\over2\pi i}\int_{\mathcal C}\zeta(s,M_K){x^s\over s}\,dsNEWLINE\]NEWLINE taken over a certain curve \(\mathcal C\) surrounding the pole at \(s=1\), which approximates \(M_K(x)\) with an error of the order \(O(x\exp(-a_K\sqrt{\log x}))\) with positive \(a_K\). Under GRH the bound NEWLINE\[NEWLINEE_K(x)=M_K(x)-T_K(x)=O\left(x^{1/2}\log^{D_K+1}x\right)NEWLINE\]NEWLINE was obtained.NEWLINENEWLINEOn the other hand it was shown by the author and \textit{A. Perelli} Am. J. Math. 120, No. 2, 289--303 (1998; Zbl 0905.11036)] that one has NEWLINE\[NEWLINEE_K(x)=\Omega_{\pm}\left(x^{1/2-\varepsilon}\right)NEWLINE\]NEWLINE for every \(\varepsilon>0\), and now the author proves NEWLINE\[NEWLINEE_K(x)=\Omega_{\pm}\left({x^{1/2}\over\log^{B_K}x}\right)NEWLINE\]NEWLINE for certain \(B_K>0\). He shows also that if the Nonvanishing Conjecture for entire functions of the Selberg class is true, then NEWLINE\[NEWLINEE_K(x)=\Omega_{\pm}\left({x^{1/2}(\log\log x)^{D_K-1}\over\log x}\right).NEWLINE\]NEWLINENEWLINENEWLINEThese results are corollaries of theorems concerning the existence of large oscillations of the error term \(E_K(x)\).