Supplementary Tables for “Numerical Results on Class Groups of Imaginary Quadratic Fields”

We present supplemental tables and additional data that extend that presented in [7]. Data corresponding to all the conjectures mentioned in [7] are included, and all tables are complete, including previously published results. In addition, two corrections to the data in [7] are included: – Originally, we only listed first occurrences of p-Sylow subgroups for primes p ≤ 173. In this paper, we present the entire list, for primes p ≤ 389. See Table 7. – When listing the first ∆ needing prime ideals of norm up to p, we pointed out an anomaly in the data at p = 181. Subsequent analysis has shown this to be a bug in our statistics gathering program. The data no longer contains any anomalies of this sort. See Table 15. Bounds on L(1, χ) There has been significant interest [2, 3, 6, 11] in the extreme values of L(1, χ∆) due to the relationship between it and the class number h∆. This can be seen in the analytic class number formula, L(1, χ∆) = h∆π where extreme values of L(1, χ∆) correspond to extreme values of h∆. In [10], Littlewood developed bounds on L(1, χ∆), namely that under the ERH, {1 + o(1)}(c1 log log ∆) −1 < L(1, χ∆) < {1 + o(1)}c2 log log(∆) , (0.1) where c1 and c2 are defined as follows: c1 = 12e γ /π 2 and c2 = 2e γ when 2 ∤ ∆ c1 = 8e γ /π 2 and c2 = e γ when 2 | ∆. ⋆ All three authors are supported in part by NSERC of Canada. In [11], Shanks investigated Littlewood’s bounds, and defined two values he termed the upper and lower Littlewood indices ULI = L(1, χ∆)/(c2 log log ∆) LLI = L(1, χ∆)c1 log log ∆. These indices effectively ignore the o(1) given in Littlewood’s bounds. We would expect extreme values of the LLI and the ULI to approach 1. Finally, as in [11], we define the function

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.