Comparison of algorithms to calculate quadratic irregularity of prime numbers
Joshua Holden
TL;DR
This work analyzes two Siegel-based approaches for computing the quadratic irregularity of primes via the zeta-values $\zeta_{D}(1-2m)$ in real quadratic fields. It derives and contrasts Siegel's elementary formula $\zeta_{D}(1-2m)=\frac{B_{2m}}{4m^{2}}D^{2m-1}\sum_{j=1}^{D}\chi(j)B_{2m}(j/D)$ and the modular-forms-derived formula $\zeta_{D}(1-2m)=-4c_{4m}^{-1}\sum_{l=1}^{r}c_{4m,l}s_{l}^{D}(2m)$, detailing their computational costs under naive, fast, and constant-time multiplication. The paper introduces an efficient Horner/Brent-based method to compute Bernoulli-related components for the first formula and presents a rearranged evaluation to reduce calls to $e_{2m-1}(n)$ in the second, highlighting trade-offs between speed in $M$ and scalability in $D$. It also outlines the computation of the coefficients $c_{4m,l}$ from modular form data, with storage and asymptotic-time considerations, and discusses practical memory constraints and potential optimizations for cryptographic applications. Overall, the results guide the choice of algorithm based on the target ranges of $M$ and $D$ and indicate avenues for further improvement, including modular techniques and caching strategies.
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 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 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.