Irregularity of prime numbers over real quadratic fields

  • Joshua Holden
Conference paper

DOI: 10.1007/BFb0054884

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

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 ko, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k0-regular primes and the index of k0-irregularity, we discuss how to compute these indices when k0 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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Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Joshua Holden
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of Massachusetts at AmherstAmherstUSA

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