Ω‐estimates related to irreducible algebraic integers |
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Authors: | Jerzy Kaczorowski |
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Affiliation: | Adam Mickiewicz University, Faculty of Mathematics and Computer Science, ul. Umultowska 87, 61‐614 Poznań, Poland |
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Abstract: | We study large values of the remainder term EK (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that EK (x) = Ω± (√(x)(log x)) for certain positive constant BK, improving in that way the previously best known estimate EK (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 that EK (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) |
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Keywords: | Factorization in algebraic number fields irreducible algebraic integers oscillations of arithmetic error terms omega theorems changes of sign Selberg class non‐vanishing on the 1‐line prime number theorem in the Selberg class Normality Conjecture Selberg Orthonrmality Conjecture |
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