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arxiv: 1906.09718 · v2 · pith:C2325H4Ynew · submitted 2019-06-24 · 🧮 math.NT

Lower bound for class number of certain real quadratic fields

Mohit Mishra This is my paper

Pith reviewed 2026-05-25 17:40 UTC · model grok-4.3

classification 🧮 math.NT
keywords class numberreal quadratic fieldsDedekind zeta functionChowla conjectureYokoi conjectureclass group
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The pith

The class number of the real quadratic field Q(sqrt(n^2 + r)) for r=1 or 4 admits an explicit lower bound equivalent to a condition on the Dedekind zeta function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives an explicit lower bound for the class number h(n^2 + r) where r equals 1 or 4 in real quadratic fields. A sympathetic reader would care because class numbers measure the failure of unique factorization in these rings, and lower bounds help classify fields with small class numbers. The bound is tied to a special value of the Dedekind zeta function via an equivalence criterion. This reduces the scope of Chowla and Yokoi's conjectures to smaller families of fields. It also supplies criteria for when the class group has prime power order and is cyclic.

Core claim

We give an explicit lower bound for h(n²+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.

What carries the argument

The analytic class-number formula combined with the functional equation of the Dedekind zeta function.

If this is right

  • The real quadratic families considered in Chowla and Yokoi's conjecture reduce to comparatively small subfamilies.
  • Equivalent criteria exist for an alternate proof of both conjectures.
  • Criteria are obtained for the class group of prime power order to be cyclic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Verifying the zeta-function condition for concrete n could determine exactly when the lower bound is achieved.
  • The reduced subfamilies might make it feasible to test the conjectures by direct computation on fewer cases.

Load-bearing premise

The analytic class-number formula and the functional equation of the Dedekind zeta function hold for these quadratic fields.

What would settle it

A computation for some integer n showing that h(n^2 + 1) falls below the paper's explicit lower bound, or that the special value of the Dedekind zeta function fails to match the equality case.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper claims to derive an explicit lower bound for the class number h(n² + r) of the real quadratic field Q(√d) with d = n² + r (r = 1 or 4), together with an equivalent criterion for equality in terms of a special value of the Dedekind zeta function ζ_K. It applies the bound to reduce the families appearing in the Chowla and Yokoi conjectures to smaller subfamilies, supplies an alternate criterion for proving those conjectures, and gives criteria under which the class group of prime-power order is cyclic.

Significance. If the explicit lower bound and the zeta-value equivalence are correctly derived from the analytic class-number formula and the functional equation, the results would be useful for reducing infinite families in class-number conjectures to finite verification and for supplying computable criteria involving L(1, χ_d). The manuscript supplies explicit bounds and equivalence statements rather than asymptotic statements, which strengthens its potential utility.

minor comments (4)
  1. The abstract states that the lower bound and equivalence are obtained but does not indicate the precise form of the bound or the exact special value (e.g., at s=0 or s=1) used in the criterion; this should be stated explicitly in the introduction or §2.
  2. Notation for the regulator R and the fundamental unit should be fixed consistently when the unit is taken to be n + √d; clarify whether this is always fundamental for the families considered.
  3. The reduction of Chowla–Yokoi families to a “comparatively small subfamily” is asserted without a quantitative statement of the size of the remaining cases; add a remark or table indicating the range of n that must still be checked.
  4. The criteria for the class group to be cyclic when its order is a prime power are stated as consequences but the logical steps linking the lower bound to cyclicity are not summarized; a short outline in the final section would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its potential utility for reducing families in the Chowla and Yokoi conjectures, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard external theorems

full rationale

The paper's central claims—an explicit lower bound for h(n²+r) (r=1,4) and an equivalence criterion phrased via the special value of the Dedekind zeta function—are obtained directly from the analytic class-number formula hR = sqrt(d) Res_{s=1} ζ_K(s) together with the functional equation relating the residue at s=1 to L(1,χ_d). Both the class-number formula and the functional equation are unconditional, externally established theorems in algebraic number theory; they are invoked rather than re-derived inside the paper. The regulator for the family d = n² + r is expressed in closed form via the fundamental unit n + sqrt(d), after which the inequality follows by comparison with the residue term. No parameter is fitted to the target class numbers, no self-citation supplies a uniqueness theorem or ansatz, and the equivalence criterion is simply a rearrangement of the same standard formula. Consequently the derivation chain contains no self-definitional, fitted-input, or self-citation-load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard analytic class-number formula and functional equation for Dedekind zeta functions of quadratic fields; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Analytic class-number formula and functional equation for the Dedekind zeta function of a real quadratic field
    Invoked to obtain both the lower bound and the equivalent criterion stated in the abstract.

pith-pipeline@v0.9.0 · 5634 in / 1274 out tokens · 29755 ms · 2026-05-25T17:40:26.163672+00:00 · methodology

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Reference graph

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