First-hit analysis of algorithms for computing quadratic irregularity

Abstract:

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo $p$ for every $p$. This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if $\mathbf {Q}(\sqrt {D})$ is a real quadratic field, then the values of the zeta function $\zeta _{D}(1-2m)=\zeta _{\mathbf {Q}(\sqrt {D})}(1-2m)$ at negative odd integers are also distributed as expected modulo $p$ for any $p$. We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.