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Ω‐estimates related to irreducible algebraic integers
Author(s) -
Kaczorowski Jerzy
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710158
Subject(s) - mathematics , algebraic number , pure mathematics , algebra over a field , mathematical analysis
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)