Distribution of Values of Real Quadratic Zeta Functions
Joshua Holden
TL;DR
The paper addresses whether zeta-values for real quadratic fields, via $ζ_D(s)=ζ(s)L(s,χ)$, exhibit uniform distribution modulo odd primes $p$ analogous to the Bernoulli-number conjecture. It introduces the notions of $k$-regular and $χ$-regular primes and tests a product-distribution model for the numerators of $L(1-2m,χ)$ using chi-squared tests across many discriminants. Findings indicate that aggregated totals can significantly deviate from predictions while per-discriminant averages often align, suggesting averaging effects or additional structure in the $D$-variation. The work highlights algorithmic advances for fixed $m$ and varying $D$, reports mixed empirical support for Siegel-type predictions, and outlines future conjectures and directions with potential links to Fermat, Vandiver, and Iwasawa-type invariants.
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 evenly distributed modulo p for every p. This is the basis of a well-known heuristic, given by Siegel, for estimating the frequency of irregular primes. So far, analyses have shown that if Q(\sqrt{D}) is a real quadratic field, then the values of the zeta function ζ_{D}(1-2m)=ζ_{Q(\sqrt{D})}(1-2m) at negative odd integers are also distributed as expected modulo p for any p. However, it has proven to be very computationally intensive to calculate these numbers for large values of m. In this paper, we present the alternative of computing ζ_{D}(1-2m) for a fixed value of D and a large number of different m.