Kronecker's class number sums

DefinitionReduced forms of a given discriminant#✓fully proved

For an integer \(d\), \(R(d)\) is the finite set of integer triples \((a, b, c)\) such that the binary quadratic form \(a x^2 + b x y + c y^2\) is Gauss reduced (that is, \(a > 0\), \(b^2 - 4ac < 0\), \(|b| \le a \le c\), and \(b \ge 0\) whenever \(|b| = a\) or \(a = c\)) and has discriminant \(b^2 - 4ac = d\). In particular \(R(d)\) is empty when \(d \ge 0\).

The set is computed by a finite search: a reduced form satisfies \(-d = 4ac - b^2 \ge 3a^2\), so \(1 \le a \le \lfloor \sqrt{-d} \rfloor\) and \(|b| \le a\), and \(c = (b^2 - d) / (4a)\) is determined by \(a\) and \(b\).

public def reducedFormsOfDiscriminant (d : ℤ) : Finset (ℤ × ℤ × ℤ) :=
  ((Finset.Icc (1 : ℤ) (Nat.sqrt (-d).toNat)).biUnion fun a =>
      (Finset.Icc (-a) a).image fun b => (a, b, (b ^ 2 - d) / (4 * a))).filter fun q =>
    BinaryQuadraticForm.IsGaussReduced q.1 q.2.1 q.2.2 ∧ discrim q.1 q.2.1 q.2.2 = d

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Section 1.

Uses 1 · Used by 4

Statement usesGauss-reduced binary quadratic forms

Used byGauss class number of a discriminant · Kronecker's sum \(F_e(n)\) · Kronecker's sum \(F_o(n)\) · Unweighted Hurwitz class number

DefinitionGauss class number of a discriminant#✓fully proved

For an integer \(d\), the Gauss class number \(h(d)\) is the number of integer triples \((a, b, c)\) with \(b^2 - 4ac = d\) for which the binary quadratic form \(ax^2 + bxy + cy^2\) is positive definite (\(a > 0\) and \(d < 0\)), reduced (\(|b| \le a \le c\), with \(b \ge 0\) whenever \(|b| = a\) or \(a = c\)) and primitive (\(\gcd(a, b, c) = 1\)).

public def gaussClassNumber (d : ℤ) : ℕ :=
  ((reducedFormsOfDiscriminant d).filter fun q =>
    Int.gcd (Int.gcd q.1 q.2.1 : ℤ) q.2.2 = 1).card

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Section 1.

Uses 1 · Used by 5

Statement usesReduced forms of a given discriminant

Used byHua's class number bound · Class numbers of the discriminants \(-3\), \(-4\) and \(-12\) · Aygin–Williams evaluation of Kronecker's sums · Gauss's three-square formula after Aygin and Williams · Unweighted Hurwitz class numbers via Gauss class numbers

DefinitionUnweighted Hurwitz class number#✓fully proved

For an integer \(d\), the unweighted Hurwitz class number \(H(d)\) is the number of elements of the set \(R(d)\) of reduced forms of discriminant \(d\), that is, the number of integer triples \((a, b, c)\), primitive or not, for which \(a x^2 + b x y + c y^2\) is a Gauss-reduced binary quadratic form of discriminant \(b^2 - 4ac = d\). Every reduced form is counted with weight \(1\); in particular \(H(d) = 0\) when \(d \ge 0\).

This differs from the weighted Hurwitz class number, which counts forms equivalent to a multiple of \(x^2 + y^2\) or of \(x^2 + x y + y^2\) with weight \(1/2\) or \(1/3\) (the paper remarks that the literature uses several conventions).

public def unweightedHurwitzClassNumber (d : ℤ) : ℕ :=
  (reducedFormsOfDiscriminant d).card

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, equation (1).

Uses 1 · Used by 2

Statement usesReduced forms of a given discriminant

Used byClass numbers of the discriminants \(-3\), \(-4\) and \(-12\) · Unweighted Hurwitz class numbers via Gauss class numbers

DefinitionWeight of a reduced binary quadratic form#✓fully proved

The weight of an integer triple \((a, b, c)\), standing for the form \(ax^2 + bxy + cy^2\), is \(1/2\) if \(a = c\) and \(b = 0\), \(1/3\) if \(a = b = c \ne 0\), and \(1\) otherwise.

On reduced forms the weight is \(1/2\) for \(a(x^2 + y^2)\), \(1/3\) for \(a(x^2 + xy + y^2)\), and \(1\) for every other form.

public def reducedFormWeight (q : ℤ × ℤ × ℤ) : ℚ :=
  if q.1 = q.2.2 ∧ q.2.1 = 0 then 1 / 2
  else if q.1 = q.2.1 ∧ q.2.1 = q.2.2 then 1 / 3 else 1

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, equation (3).

Used by 2

Used byKronecker's sum \(F_e(n)\) · Kronecker's sum \(F_o(n)\)

DefinitionKronecker's sum \(F_o(n)\)#✓fully proved

For a natural number \(n\), \(F_o(n)\) is the sum of the weights of the reduced positive-definite integral binary quadratic forms \(ax^2 + bxy + cy^2\) of discriminant \(b^2 - 4ac = -4n\) in which \(a\) or \(c\) is odd. Here a form is reduced when \(|b| \le a \le c\), with \(b \ge 0\) whenever \(|b| = a\) or \(a = c\), and its weight is \(1/2\) if \(a = c\) and \(b = 0\), \(1/3\) if \(a = b = c\), and \(1\) otherwise.

public def kroneckerFOdd (n : ℕ) : ℚ :=
  ∑ q ∈ (reducedFormsOfDiscriminant (-4 * n)).filter (fun q => Odd q.1 ∨ Odd q.2.2),
    reducedFormWeight q

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, equation (4).

Uses 2 · Used by 2

Statement usesWeight of a reduced binary quadratic form · Reduced forms of a given discriminant

Used byKronecker's sums at \(n = 1\) and \(n = 3\) · Aygin–Williams evaluation of Kronecker's sums

DefinitionKronecker's sum \(F_e(n)\)#✓fully proved

For a natural number \(n\), \(F_e(n)\) is the sum of the weights of the reduced positive-definite integral binary quadratic forms \(ax^2 + bxy + cy^2\) of discriminant \(b^2 - 4ac = -4n\) in which \(a\) and \(c\) are both even. Here a form is reduced when \(|b| \le a \le c\), with \(b \ge 0\) whenever \(|b| = a\) or \(a = c\), and its weight is \(1/2\) if \(a = c\) and \(b = 0\), \(1/3\) if \(a = b = c\), and \(1\) otherwise.

public def kroneckerFEven (n : ℕ) : ℚ :=
  ∑ q ∈ (reducedFormsOfDiscriminant (-4 * n)).filter (fun q => Even q.1 ∧ Even q.2.2),
    reducedFormWeight q

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, equation (5).

Uses 2 · Used by 2

Statement usesWeight of a reduced binary quadratic form · Reduced forms of a given discriminant

Used byKronecker's sums at \(n = 1\) and \(n = 3\) · Aygin–Williams evaluation of Kronecker's sums

DefinitionThe discriminant \(d(\alpha, g)\) attached to \(n = 2^\alpha g h^2\)#✓fully proved

For natural numbers \(\alpha\) and \(g\), let \(d(\alpha, g) = -8g\) if \(\alpha\) is odd; if \(\alpha\) is even, let \(d(\alpha, g) = -4g\) when \(g \equiv 1 \pmod 4\) and \(d(\alpha, g) = -g\) otherwise.

In the paper \(n = 2^\alpha g h^2\) with \(g\) and \(h\) odd and \(g\) squarefree, so the last case is \(g \equiv 3 \pmod 4\); the paper writes \(d(n)\) for \(d(\alpha, g)\), the fundamental discriminant of \(\mathbb{Q}(\sqrt{-n})\).

public def kroneckerDiscriminant (α g : ℕ) : ℤ :=
  if Even α then (if g % 4 = 1 then -4 * g else -g) else -8 * g

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Section 1, after equation (17).

Used by 4

Used bySmall values of the discriminant \(d(\alpha, g)\) and the multiplier \(\ell(d, h)\) · Aygin–Williams evaluation of Kronecker's sums · Gauss's three-square formula after Aygin and Williams · Unweighted Hurwitz class numbers via Gauss class numbers

DefinitionKronecker's class-number multiplier#✓fully proved

For an integer \(d\) and a natural number \(h \ge 1\), \(\ell(d, h)\) is the product, over the primes \(p\) dividing \(h\), of

\[\sum_{i=0}^{v} p^i - \left(\frac{d}{p}\right) \sum_{i=0}^{v-1} p^i,\]

where \(v\) is the exponent of \(p\) in \(h\) and \(\left(\frac{d}{p}\right)\) is the Jacobi symbol; in particular \(\ell(d, 1) = 1\). Also \(\ell(d, 0) = 1\).

In the paper \(h\) is odd, so each \(p\) is an odd prime, the symbol is the Legendre symbol, and the factor equals \(\frac{p^{v+1} - 1}{p - 1} - \left(\frac{d}{p}\right) \frac{p^v - 1}{p - 1}\).

public def kroneckerClassMultiplier (d : ℤ) (h : ℕ) : ℤ :=
  ∏ p ∈ h.primeFactors,
    ((∑ i ∈ Finset.range (h.factorization p + 1), (p : ℤ) ^ i) -
      jacobiSym d p * ∑ i ∈ Finset.range (h.factorization p), (p : ℤ) ^ i)

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Section 1, definition of ℓ(n).

Used by 4

Used bySmall values of the discriminant \(d(\alpha, g)\) and the multiplier \(\ell(d, h)\) · Aygin–Williams evaluation of Kronecker's sums · Gauss's three-square formula after Aygin and Williams · Unweighted Hurwitz class numbers via Gauss class numbers

TheoremClass numbers of the discriminants \(-3\), \(-4\) and \(-12\)#✓fully proved

The Gauss class numbers satisfy \(h(-3) = h(-4) = h(-12) = 1\), and the unweighted Hurwitz class number satisfies \(H(-12) = 2\).

The two reduced forms of discriminant \(-12\) are \(x^2 + 3y^2\) and the imprimitive form \(2x^2 + 2xy + 2y^2\).

public theorem gaussClassNumber_small :
    gaussClassNumber (-3) = 1 ∧ gaussClassNumber (-4) = 1 ∧ gaussClassNumber (-12) = 1 ∧
      unweightedHurwitzClassNumber (-12) = 2

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Section 1.

Uses 2

Statement usesGauss class number of a discriminant · Unweighted Hurwitz class number

TheoremKronecker's sums at \(n = 1\) and \(n = 3\)#✓fully proved

Kronecker's sums \(F_o\) and \(F_e\) satisfy \(F_o(1) = 1/2\), \(F_e(1) = 0\), \(F_o(3) = 1\) and \(F_e(3) = 1/3\).

public theorem kroneckerF_small :
    kroneckerFOdd 1 = 1 / 2 ∧ kroneckerFEven 1 = 0 ∧ kroneckerFOdd 3 = 1 ∧
      kroneckerFEven 3 = 1 / 3

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, equations (4) and (5).

Uses 2

Statement usesKronecker's sum \(F_e(n)\) · Kronecker's sum \(F_o(n)\)

TheoremSmall values of the discriminant \(d(\alpha, g)\) and the multiplier \(\ell(d, h)\)#✓fully proved

The discriminant attached to \(n = 2^\alpha g h^2\) satisfies \(d(0, 1) = -4\), \(d(0, 3) = -3\) and \(d(1, 1) = -8\), and Kronecker's class-number multiplier satisfies \(\ell(-4, 1) = 1\) and \(\ell(-4, 3) = 5\).

public theorem kroneckerDiscriminant_kroneckerClassMultiplier_small :
    kroneckerDiscriminant 0 1 = -4 ∧ kroneckerDiscriminant 0 3 = -3 ∧
      kroneckerDiscriminant 1 1 = -8 ∧ kroneckerClassMultiplier (-4) 1 = 1 ∧
      kroneckerClassMultiplier (-4) 3 = 5

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Section 1.

Uses 2

Statement usesKronecker's class-number multiplier · The discriminant \(d(\alpha, g)\) attached to \(n = 2^\alpha g h^2\)

TheoremGauss's three-square formula after Aygin and Williams#●ready to prove

Let \(n\), \(\alpha\), \(g\), \(h\) be natural numbers with \(n = 2^\alpha g h^2\), where \(g\) and \(h\) are odd and \(g\) is squarefree. Let \(d\) be the Kronecker discriminant of \((\alpha, g)\), namely \(d = -4g\) if \(\alpha\) is even and \(g \equiv 1 \pmod 4\), \(d = -g\) if \(\alpha\) is even and \(g \equiv 3 \pmod 4\), and \(d = -8g\) if \(\alpha\) is odd. Put \(L = \ell(d, h) \, h(d)\), where \(\ell(d, h)\) is the Kronecker class multiplier and \(h(d)\) is the Gauss class number of \(d\), and let \(r_3(n)\) be the number of ordered triples \((x, y, z)\) of integers with \(x^2 + y^2 + z^2 = n\). Then: if \(\alpha\) is even and \(g = 1\), then \(r_3(n) = 6L\); if \(\alpha\) is even and \(g = 3\), then \(r_3(n) = 8L\); if \(\alpha\) is even, \(g \equiv 1 \pmod 4\) and \(g \ne 1\), then \(r_3(n) = 12L\); if \(\alpha\) is even, \(g \equiv 3 \pmod 8\) and \(g \ne 3\), then \(r_3(n) = 24L\); if \(\alpha\) is even and \(g \equiv 7 \pmod 8\), then \(r_3(n) = 0\); and if \(\alpha\) is odd, then \(r_3(n) = 12L\).

public theorem_wanted threeSquareRepresentationCount_eq_kronecker_classNumber
    (n α g h : ℕ) (hn : n = 2 ^ α * g * h ^ 2) (hg : Odd g) (hgsq : Squarefree g)
    (hh : Odd h) :
    let L : ℤ := kroneckerClassMultiplier (kroneckerDiscriminant α g) h *
      gaussClassNumber (kroneckerDiscriminant α g)
    let r : ℤ := Nat.threeSquareRepresentationCount n
    (Even α → g = 1 → r = 6 * L) ∧
    (Even α → g = 3 → r = 8 * L) ∧
    (Even α → g % 4 = 1 → g ≠ 1 → r = 12 * L) ∧
    (Even α → g % 8 = 3 → g ≠ 3 → r = 24 * L) ∧
    (Even α → g % 8 = 7 → r = 0) ∧
    (Odd α → r = 12 * L)

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Theorem 1.

Uses 4

Statement usesGauss class number of a discriminant · Kronecker's class-number multiplier · The discriminant \(d(\alpha, g)\) attached to \(n = 2^\alpha g h^2\) · Number of representations as a sum of three squares

TheoremAygin–Williams evaluation of Kronecker's sums#●ready to prove

Let \(n = 2^\alpha g h^2\) with \(g\) and \(h\) odd natural numbers and \(g\) squarefree. Let \(d = -4g\) if \(\alpha\) is even and \(g \equiv 1 \pmod 4\), \(d = -g\) if \(\alpha\) is even and \(g \equiv 3 \pmod 4\), and \(d = -8g\) if \(\alpha\) is odd. Let \(L = \ell(d, h) \cdot h(d)\), where \(\ell(d, h)\) is Kronecker's class-number multiplier and \(h(d)\) is the Gauss class number of \(d\), and for even \(\alpha\) let \(t = 2^{\alpha/2}\). Then Kronecker's sums \(F_o\) and \(F_e\) satisfy:

• if \(\alpha\) is even and \(g = 1\), then \(F_o(n) = \frac{t}{2} L\) and \(F_e(n) = \left(\frac{t}{2} - \frac{1}{2}\right) L\);
• if \(\alpha\) is even and \(g = 3\), then \(F_o(n) = tL\) and \(F_e(n) = \left(t - \frac{2}{3}\right) L\);
• if \(\alpha\) is even, \(g \equiv 1 \pmod 4\) and \(g \ne 1\), then \(F_o(n) = tL\) and \(F_e(n) = (t - 1) L\);
• if \(\alpha\) is even, \(g \equiv 3 \pmod 8\) and \(g \ne 3\), then \(F_o(n) = 3tL\) and \(F_e(n) = (3t - 2) L\);
• if \(\alpha\) is even and \(g \equiv 7 \pmod 8\), then \(F_o(n) = F_e(n) = tL\);
• if \(\alpha\) is odd, then \(F_o(n) = 2^{(\alpha - 1)/2} L\) and \(F_e(n) = \left(2^{(\alpha - 1)/2} - 1\right) L\).

MetaMathlibExt.kroneckerFOdd_kroneckerFEven_eq_classNumber · 19 lines

public theorem_wanted kroneckerFOdd_kroneckerFEven_eq_classNumber
    (n α g h : ℕ) (hn : n = 2 ^ α * g * h ^ 2) (hg : Odd g) (hgsq : Squarefree g)
    (hh : Odd h) :
    let L : ℚ := (kroneckerClassMultiplier (kroneckerDiscriminant α g) h : ℚ) *
      gaussClassNumber (kroneckerDiscriminant α g)
    let t : ℚ := 2 ^ (α / 2)
    (Even α → g = 1 →
      kroneckerFOdd n = t / 2 * L ∧ kroneckerFEven n = (t / 2 - 1 / 2) * L) ∧
    (Even α → g = 3 →
      kroneckerFOdd n = t * L ∧ kroneckerFEven n = (t - 2 / 3) * L) ∧
    (Even α → g % 4 = 1 → g ≠ 1 →
      kroneckerFOdd n = t * L ∧ kroneckerFEven n = (t - 1) * L) ∧
    (Even α → g % 8 = 3 → g ≠ 3 →
      kroneckerFOdd n = 3 * t * L ∧ kroneckerFEven n = (3 * t - 2) * L) ∧
    (Even α → g % 8 = 7 →
      kroneckerFOdd n = t * L ∧ kroneckerFEven n = t * L) ∧
    (Odd α →
      kroneckerFOdd n = (2 : ℚ) ^ ((α - 1) / 2) * L ∧
        kroneckerFEven n = ((2 : ℚ) ^ ((α - 1) / 2) - 1) * L)

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Theorem 2.

Uses 5

Statement usesGauss class number of a discriminant · Kronecker's class-number multiplier · The discriminant \(d(\alpha, g)\) attached to \(n = 2^\alpha g h^2\) · Kronecker's sum \(F_e(n)\) · Kronecker's sum \(F_o(n)\)

TheoremUnweighted Hurwitz class numbers via Gauss class numbers#●ready to prove

Let \(n\), \(\alpha\), \(g\), \(h\) be natural numbers with \(n = 2^\alpha g h^2\), where \(g\) and \(h\) are odd and \(g\) is squarefree. Let \(d\) be the Kronecker discriminant of \((\alpha, g)\), namely \(d = -4g\) if \(\alpha\) is even and \(g \equiv 1 \pmod 4\), \(d = -g\) if \(\alpha\) is even and \(g \equiv 3 \pmod 4\), and \(d = -8g\) if \(\alpha\) is odd. Put \(L = \ell(d, h) \, h(d)\), where \(\ell(d, h)\) is the Kronecker class multiplier and \(h(d)\) is the Gauss class number of \(d\), and let \(H(-4n)\) be the unweighted Hurwitz class number of \(-4n\). Then, with \(\alpha / 2\) and \((\alpha + 1) / 2\) exact quotients in the cases where they occur: if \(\alpha\) is even and \(g = 1\), then \(H(-4n) = (2^{\alpha/2} - 1/2) L + 1/2\); if \(\alpha\) is even and \(g = 3\), then \(H(-4n) = (2^{\alpha/2 + 1} - 2/3) L + 2/3\); if \(\alpha\) is even, \(g \ne 1\) and \(g \equiv 1 \pmod 4\), then \(H(-4n) = (2^{\alpha/2 + 1} - 1) L\); if \(\alpha\) is even, \(g \ne 3\) and \(g \equiv 3 \pmod 8\), then \(H(-4n) = (3 \cdot 2^{\alpha/2 + 1} - 2) L\); if \(\alpha\) is even and \(g \equiv 7 \pmod 8\), then \(H(-4n) = 2^{\alpha/2 + 1} L\); and if \(\alpha\) is odd, then \(H(-4n) = (2^{(\alpha + 1)/2} - 1) L\).

public theorem_wanted unweightedHurwitzClassNumber_neg_four_mul_eq_classNumber
    (n α g h : ℕ) (hn : n = 2 ^ α * g * h ^ 2) (hg : Odd g) (hgsq : Squarefree g)
    (hh : Odd h) :
    let L : ℚ := (kroneckerClassMultiplier (kroneckerDiscriminant α g) h : ℚ) *
      gaussClassNumber (kroneckerDiscriminant α g)
    let H : ℚ := unweightedHurwitzClassNumber (-4 * n)
    (Even α → g = 1 → H = ((2 : ℚ) ^ (α / 2) - 1 / 2) * L + 1 / 2) ∧
    (Even α → g = 3 → H = ((2 : ℚ) ^ (α / 2 + 1) - 2 / 3) * L + 2 / 3) ∧
    (Even α → g ≠ 1 → g % 4 = 1 → H = ((2 : ℚ) ^ (α / 2 + 1) - 1) * L) ∧
    (Even α → g ≠ 3 → g % 8 = 3 → H = (3 * (2 : ℚ) ^ (α / 2 + 1) - 2) * L) ∧
    (Even α → g % 8 = 7 → H = (2 : ℚ) ^ (α / 2 + 1) * L) ∧
    (Odd α → H = ((2 : ℚ) ^ ((α + 1) / 2) - 1) * L)

Sources

• Zafer Selcuk Aygin and Kenneth S. Williams, A Note on the Evaluation of Two Arithmetic Functions of Kronecker, INTEGERS 26 (2026), Article A29, Theorem 3.

Uses 4

Statement usesGauss class number of a discriminant · Kronecker's class-number multiplier · The discriminant \(d(\alpha, g)\) attached to \(n = 2^\alpha g h^2\) · Unweighted Hurwitz class number

11 fully proved3 ready to prove

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