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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Independence of Hecke zeta functions of finite order over normal fields

Author: Maciej Radziejewski
Journal: Trans. Amer. Math. Soc. 359 (2007), 2383-2394
MSC (2000): Primary 11N64
Published electronically: December 15, 2006
MathSciNet review: 2276625
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Abstract: We study oscillations of the remainder term corresponding to the counting functions of the sets of elements with unique factorization length in semigroups of algebraic numbers such as the semigroup of algebraic integers or totally positive algebraic integers in a given normal field $ K$. The results imply existence of oscillations when the exponent of the class group of the semigroup in question is sufficiently large depending on the field's degree. In particular, when $ K$ is a quadratic field or a normal cubic field oscillations exist whenever the class group is not isomorphic to $ C_2^a \oplus C_3^b \oplus C_4^c$ for nonnegative integers $ a, b, c$. The main part of this study is concerned with the problem of multiplicative independence of Hecke zeta functions. We also show that there are infinitely many fields whose Dedekind zeta function has infinitely many nontrivial multiple zeros.

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Additional Information

Maciej Radziejewski
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

Keywords: Functional independence, Selberg class, oscillations, error terms, Mellin transforms, Hilbert semigroup, factorizations of distinct lengths
Received by editor(s): November 6, 2004
Received by editor(s) in revised form: April 16, 2005
Published electronically: December 15, 2006
Additional Notes: This work was supported by the Foundation for Polish Science and by the Polish Research Committee (KBN grant No. 1P03A00826).
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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