Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E K ( x ) in the asymptotic formula for the number of irreducible integers in an algebraic number field K . We show that E K ( x ) = Ω ± (√( x )(log x )   – B   K) for certain positive constant B K , improving in that way the previously best known estimateE K ( x ) = Ω ± ( x (1/2)‐ ε )for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L ‐function from the Selberg class vanishes on the vertical line σ = 1, we show thatE K ( x ) = Ω ± (√( x )(log log x ) D ( K )‐1 (log x ) ‐1 ),supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)