Moments of Dedekind zeta coefficients over sums of six squares

TheoremFourth moment of the \(L\)-function modulo \(4\)#●ready to prove

Let \(\chi\) be a nontrivial Dirichlet character modulo \(4\) and let \(\varepsilon > 0\). Then there is a real constant \(C\) such that for every real \(T \ge 1\), \(\int_1^T |L(1/2 + it, \chi)|^4 \, dt \le C\, T^{1 + \varepsilon}\).

public theorem_wanted dirichletLFunction_mod_four_fourth_moment_le {ε : ℝ} (hε : 0 < ε)
    (χ : DirichletCharacter ℂ 4) (hχ : χ ≠ 1) :
    ∃ C : ℝ, ∀ T : ℝ, 1 ≤ T →
      ∫ t in (1 : ℝ)..T, ‖χ.LFunction (1 / 2 + t * I)‖ ^ 4 ≤ C * T ^ (1 + ε)

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Lemma 5, third display.

DefinitionCoefficients of the Dedekind zeta function#✓fully proved

For a number field \(K\) with ring of integers \(\mathcal{O}_K\) and a natural number \(n\), \(a_K(n)\) is the number of ideals of \(\mathcal{O}_K\) of absolute norm \(n\). These are the coefficients of \(\zeta_K(s) = \sum_{n \ge 1} a_K(n)\, n^{-s}\).

noncomputable def dedekindZetaCoeff (K : Type*) [Field K] [NumberField K] (n : ℕ) : ℕ :=
  Nat.card {I : Ideal (𝓞 K) // Ideal.absNorm I = n}

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Section 1, equation (2).

Used by 3

Used byThe Dedekind zeta coefficient at \(1\) · Six-squares moments of Dedekind zeta coefficients · The six-squares moment vanishes below \(1\)

TheoremThe Dedekind zeta coefficient at \(1\)#✓fully proved

For every number field \(K\), \(a_K(1) = 1\), where \(a_K(n)\) is the number of ideals of the ring of integers of \(K\) of absolute norm \(n\): the unit ideal is the only ideal of norm \(1\).

theorem dedekindZetaCoeff_one (K : Type*) [Field K] [NumberField K] :
    dedekindZetaCoeff K 1 = 1

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Section 1, equation (2).

Uses 1

Statement usesCoefficients of the Dedekind zeta function

DefinitionSix-squares moments of Dedekind zeta coefficients#✓fully proved

For a number field \(K\), a natural number \(\ell\) and a real number \(x\), the moment is the real number \(S_\ell(x) = \sum_{1 \le n \le x} r_6(n)\, a_K(n)^\ell\), where \(n\) runs over the integers in \([1, x]\), \(a_K(n)\) is the number of ideals of the ring of integers of \(K\) of absolute norm \(n\), and \(r_6(n)\) is the coefficient of \(q^n\) in \(\theta(q)^6\), with \(\theta(q) = \sum_{m \in \mathbb{Z}} q^{m^2}\); that is, \(r_6(n)\) is the number of representations of \(n\) as a sum of six squares of integers.

noncomputable def sixSquaresDedekindMoment (K : Type*) [Field K] [NumberField K] (ℓ : ℕ)
    (x : ℝ) : ℝ :=
  ∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (sumOfSquaresRepCount 6 n : ℝ) * (dedekindZetaCoeff K n : ℝ) ^ ℓ

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Section 1, equation (4), and Section 2, equation (12).

Uses 2 · Used by 3

Statement usesCoefficients of the Dedekind zeta function · Representation numbers for sums of squares

Used byHigher six-squares moments over a non-normal cubic field · The six-squares moment vanishes below \(1\) · Second and third six-squares moments over a cubic field of negative discriminant

TheoremThe six-squares moment vanishes below \(1\)#✓fully proved

For every number field \(K\), every natural number \(\ell\) and every real \(x < 1\), the six-squares moment \(S_\ell(x) = \sum_{1 \le n \le x} r_6(n)\, a_K(n)^\ell\) equals \(0\), where \(r_6(n)\) is the number of representations of \(n\) as a sum of six squares of integers and \(a_K(n)\) is the number of ideals of the ring of integers of \(K\) of absolute norm \(n\).

theorem sixSquaresDedekindMoment_of_lt_one (K : Type*) [Field K] [NumberField K] (ℓ : ℕ)
    {x : ℝ} (hx : x < 1) : sixSquaresDedekindMoment K ℓ x = 0

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Section 1, equation (4).

Uses 3

Statement usesSix-squares moments of Dedekind zeta coefficients

Proof usesCoefficients of the Dedekind zeta function · Representation numbers for sums of squares

DefinitionThe exponent \(a_{0,\ell}\)#✓fully proved

For a natural number \(\ell\), define

\[a_{0,\ell} = 1 + \sum_{i=1}^{\lfloor \ell/2 \rfloor} \binom{\ell}{2i} C_i,\]

where \(C_i = (2i)!/(i!\,(i+1)!)\) is the \(i\)-th Catalan number.

def dedekindSixSquaresA0 (ℓ : ℕ) : ℕ :=
  1 + ∑ i ∈ Finset.Icc 1 (ℓ / 2), ℓ.choose (2 * i) * catalan i

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Theorem 2 and equation (19).

Used by 2

Used byValues of \(a_{0,\ell}\) and \(a_{3,\ell}\) for \(\ell = 3, 4, 5\) · Higher six-squares moments over a non-normal cubic field

DefinitionThe exponent \(a_{3,\ell}\)#✓fully proved

For a natural number \(\ell\), \(a_{3,\ell}\) is the sum over the integers \(1 \le i \le \lfloor (\ell - 1)/2 \rfloor\) of \(\binom{\ell}{2i+1} q_i\), where \(q_i\) is the integer part of \(4\,(2i+1)!/((i-1)!\,(i+3)!)\).

Each quotient \(4\,(2i+1)!/((i-1)!\,(i+3)!)\) is in fact an integer, and the upper limit \(\lfloor (\ell - 1)/2 \rfloor\) equals the source's \(\lceil \ell/2 - 1 \rceil\).

def dedekindSixSquaresA3 (ℓ : ℕ) : ℕ :=
  ∑ i ∈ Finset.Icc 1 ((ℓ - 1) / 2), ℓ.choose (2 * i + 1) *
    (4 * (2 * i + 1).factorial / ((i - 1).factorial * (i + 3).factorial))

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Theorem 2.

Used by 2

Used byValues of \(a_{0,\ell}\) and \(a_{3,\ell}\) for \(\ell = 3, 4, 5\) · Higher six-squares moments over a non-normal cubic field

TheoremValues of \(a_{0,\ell}\) and \(a_{3,\ell}\) for \(\ell = 3, 4, 5\)#✓fully proved

For a natural number \(\ell\) let \(a_{0,\ell} = 1 + \sum_{i=1}^{\lfloor \ell/2 \rfloor} \binom{\ell}{2i} C_i\), with \(C_i\) the \(i\)-th Catalan number, and let \(a_{3,\ell}\) be the sum over the integers \(1 \le i \le \lfloor (\ell - 1)/2 \rfloor\) of \(\binom{\ell}{2i+1}\) times the integer part of \(4\,(2i+1)!/((i-1)!\,(i+3)!)\). Then \((a_{0,3}, a_{3,3}) = (4, 1)\), \((a_{0,4}, a_{3,4}) = (9, 4)\) and \((a_{0,5}, a_{3,5}) = (21, 14)\).

theorem dedekindSixSquaresA0_A3_small :
    (dedekindSixSquaresA0 3, dedekindSixSquaresA3 3) = (4, 1) ∧
      (dedekindSixSquaresA0 4, dedekindSixSquaresA3 4) = (9, 4) ∧
      (dedekindSixSquaresA0 5, dedekindSixSquaresA3 5) = (21, 14)

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Table 1.

Uses 2

Statement usesThe exponent \(a_{0,\ell}\) · The exponent \(a_{3,\ell}\)

TheoremSecond and third six-squares moments over a cubic field of negative discriminant#●ready to prove

Let \(K\) be a number field of degree \(3\) over \(\mathbb{Q}\) with negative discriminant. For \(n \ge 1\) let \(a_K(n)\) be the number of ideals of the ring of integers of \(K\) of absolute norm \(n\) and \(r_6(n)\) the number of representations of \(n\) as a sum of six squares of integers, and for real \(x\) and natural \(\ell\) let \(S_\ell(x) = \sum_{1 \le n \le x} r_6(n)\, a_K(n)^\ell\). Then there are real constants \(c_1, c_2\) such that for every \(\varepsilon > 0\), \(S_2(x) = c_1 x^3 + c_2 x^3 \log x + O(x^{25/9 + \varepsilon})\) as \(x \to \infty\), and there is a real polynomial \(P\) of degree \(4\) such that for every \(\varepsilon > 0\), \(S_3(x) = x^3 P(\log x) + O(x^{79/27 + \varepsilon})\) as \(x \to \infty\).

A cubic field of negative discriminant is not normal over \(\mathbb{Q}\).

public theorem_wanted sixSquaresDedekindMoment_two_three (K : Type*) [Field K] [NumberField K]
    (hK : Module.finrank ℚ K = 3) (hD : discr K < 0) :
    (∃ c₁ c₂ : ℝ, ∀ ε : ℝ, 0 < ε →
      (fun x => sixSquaresDedekindMoment K 2 x - c₁ * x ^ 3 - c₂ * x ^ 3 * Real.log x) =O[Filter.atTop]
        fun x => x ^ ((25 : ℝ) / 9 + ε)) ∧
    ∃ P : Polynomial ℝ, P.natDegree = 4 ∧ ∀ ε : ℝ, 0 < ε →
      (fun x => sixSquaresDedekindMoment K 3 x - x ^ 3 * P.eval (Real.log x)) =O[Filter.atTop]
        fun x => x ^ ((79 : ℝ) / 27 + ε)

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Theorem 1.

Uses 1

Statement usesSix-squares moments of Dedekind zeta coefficients

TheoremHigher six-squares moments over a non-normal cubic field#●ready to prove

Let \(K\) be a number field of degree \(3\) over \(\mathbb{Q}\) with negative discriminant and let \(\ell \ge 4\). For \(n \ge 1\) let \(a_K(n)\) be the number of ideals of the ring of integers of \(K\) of absolute norm \(n\) and \(r_6(n)\) the number of representations of \(n\) as a sum of six squares of integers, and for real \(x\) let \(S_\ell(x) = \sum_{1 \le n \le x} r_6(n)\, a_K(n)^\ell\). Let \(a_{0,\ell} = 1 + \sum_{i=1}^{\lfloor \ell/2 \rfloor} \binom{\ell}{2i} C_i\), with \(C_i\) the \(i\)-th Catalan number, and let \(a_{3,\ell}\) be the sum over the integers \(1 \le i \le \lfloor (\ell - 1)/2 \rfloor\) of \(\binom{\ell}{2i+1}\) times the integer part of \(4\,(2i+1)!/((i-1)!\,(i+3)!)\). Then there is a real polynomial \(P\) of degree \(a_{0,\ell} + a_{3,\ell} - 1\) such that for every \(\varepsilon > 0\), \(S_\ell(x) - x^3 P(\log x) = O(x^{3 - 2/3^\ell + \varepsilon})\) as \(x \to \infty\).

Such a field is not normal over \(\mathbb{Q}\). The polynomial is called \(P_\ell\) for even \(\ell\) and \(Q_\ell\) for odd \(\ell\) in the source.

public theorem_wanted sixSquaresDedekindMoment_ge_four (K : Type*) [Field K] [NumberField K]
    (hK : Module.finrank ℚ K = 3) (hD : discr K < 0) (ℓ : ℕ) (hℓ : 4 ≤ ℓ) :
    ∃ P : Polynomial ℝ, P.natDegree = dedekindSixSquaresA0 ℓ + dedekindSixSquaresA3 ℓ - 1 ∧
      ∀ ε : ℝ, 0 < ε →
        (fun x => sixSquaresDedekindMoment K ℓ x - x ^ 3 * P.eval (Real.log x)) =O[Filter.atTop]
          fun x => x ^ (3 - 2 / (3 : ℝ) ^ ℓ + ε)

Sources

• Naveen K. Godara and Prashant Tiwari, The Higher Power Moments of the Coefficients of the Dedekind Zeta Function over a Polynomial in Six Variables, INTEGERS 25 (2025), Article A101, Theorem 2.

Uses 3

Statement usesThe exponent \(a_{0,\ell}\) · The exponent \(a_{3,\ell}\) · Six-squares moments of Dedekind zeta coefficients

7 fully proved3 ready to prove

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