Comparison of algorithms to calculate quadratic irregularity of prime numbers
Joshua Holden
在之前的工作,作者已经扩展了常规和不规则素数的概念到任意完全实数字段k_0的设置,使用负整数的zeta函数 ζ_k_0 的值作为我们的“更高的伯努利数”。 在k_0是一个真正的二次场的情况下,西格尔提出了计算这些zeta-values的两个公式:一个使用完全基本的方法,一个来自模块化形式理论。 (作者要感谢Henri Cohen建议对第二个公式进行分析。 我们简要讨论了基于这些公式的几种算法,并比较了使用它们来确定一个素数的k_0-不规则性(更普遍的是“二次不规则性”)的索引。
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 ζ_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...