Lower bound for class number of certain real quadratic fields

TL;DR

This paper establishes explicit lower bounds for the class number of certain real quadratic fields and provides criteria related to Dedekind zeta functions, aiding in the analysis of Chowla and Yokoi's conjectures.

Contribution

It introduces explicit lower bounds for class numbers of fields of the form n^2+r (r=1,4) and links these bounds to Dedekind zeta values, reducing the scope of related conjectures.

Findings

Derived explicit lower bounds for class numbers of specific quadratic fields.

Established criteria connecting class number bounds to Dedekind zeta function values.

Provided conditions for class groups of prime power order to be cyclic.

Abstract

Let $d$ be a square-free positive integer and $h(d)$ the class number of the real quadratic field $\mathbb{Q}{(\sqrt{d})}.$ In this paper we give an explicit lower bound for $h(n^2+r)$, where $r=1,4$, and also establish an equivalent criteria to attain this lower bound in terms of special value of Dedekind zeta function. Our bounds enable us to reduce the real quadratic families considered in Chowla and Yokoi's conjecture to comparatively small subfamily. We also give an equivalent criteria for having an alternate proof of both the conjectures. Also applying our results, we obtain some criteria for class group of prime power order to be cyclic.