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International Algorithmic Number Theory Symposium

ANTS 1998: Algorithmic Number Theory pp 454–462Cite as

Irregularity of prime numbers over real quadratic fields

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

Abstract

The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k 0-regular primes and the index of k 0-irregularity, we discuss how to compute these indices when k 0 is a real quadratic field. Finally, we present the results of some preliminary computations, and show that the frequency of various indices seems to agree with those predicted by a heuristic argument.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts at Amherst, 01003, Amherst, MA, USA

    Joshua Holden

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  1. Joshua Holden
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Joe P. Buhler

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© 1998 Springer-Verlag Berlin Heidelberg

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Holden, J. (1998). Irregularity of prime numbers over real quadratic fields. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054884

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  • DOI: https://doi.org/10.1007/BFb0054884

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64657-0

  • Online ISBN: 978-3-540-69113-6

  • eBook Packages: Springer Book Archive

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