Comparison of algorithms to calculate quadratic irregularity of prime numbers

Abstract

In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k_{0}, using the values of the zeta function \zeta_{k_{0}} at negative integers as our ``higher Bernoulli numbers''. In the case where k_{0} is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of k_{0}-irregularity (more generally, ``quadratic irregularity'') of a prime number.