Mean value of 2^ω(n) over squareful numbers

TheoremFine zeta factorization of the squareful \(2^{\omega(n)}\) Dirichlet series#●ready to prove

Let \(\omega(n)\) be the number of distinct prime factors of \(n\), call \(n\) squareful when \(n \ne 0\) and \(p^2 \mid n\) for every prime \(p \mid n\) (that is, \(n\) is \(2\)-full), and let \(F(s) = \sum_{n \text{ squareful}} 2^{\omega(n)} n^{-s}\). There is a sequence \(h : \mathbb{N} \to \mathbb{C}\) whose Dirichlet series \(H(s) = \sum_{n \ge 1} h(n) n^{-s}\) converges absolutely for \(\operatorname{Re} s > 1/15\), such that for every complex \(s\) with \(\operatorname{Re} s > 1/2\),

\[F(s) = \frac{\zeta(2s)^2 \zeta(3s)^2 \zeta(7s)^2 \zeta(8s)^4 \zeta(12s)^5 \zeta(13s)^{14} \zeta(14s)^5}{\zeta(4s) \zeta(5s)^2 \zeta(6s)^3 \zeta(10s)^5 \zeta(11s)^6} H(s).\]
public theorem_wanted squareful_two_pow_omega_LSeries_eq_zeta_quotient_fifteen :
    ∃ h : ℕ → ℂ, (∀ s : ℂ, 1 / 15 < s.re → LSeriesSummable h s) ∧
      ∀ s : ℂ, 1 / 2 < s.re →
        LSeries (fun n => if Nat.IsKFull 2 n then
            (2 : ℂ) ^ ArithmeticFunction.cardDistinctFactors n else 0) s =
          riemannZeta (2 * s) ^ 2 * riemannZeta (3 * s) ^ 2 * riemannZeta (7 * s) ^ 2 *
              riemannZeta (8 * s) ^ 4 * riemannZeta (12 * s) ^ 5 *
              riemannZeta (13 * s) ^ 14 * riemannZeta (14 * s) ^ 5 /
            (riemannZeta (4 * s) * riemannZeta (5 * s) ^ 2 * riemannZeta (6 * s) ^ 3 *
              riemannZeta (10 * s) ^ 5 * riemannZeta (11 * s) ^ 6) * LSeries h s
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 1.
Uses 1
Statement uses\(k\)-full numbers

TheoremCoarse zeta factorization of the squareful \(2^{\omega(n)}\) Dirichlet series#●ready to prove

Let \(\omega(n)\) be the number of distinct prime factors of \(n\), call \(n\) squareful when \(n \ne 0\) and \(p^2 \mid n\) for every prime \(p \mid n\) (that is, \(n\) is \(2\)-full), and let \(F(s) = \sum_{n \text{ squareful}} 2^{\omega(n)} n^{-s}\). There is a sequence \(h : \mathbb{N} \to \mathbb{C}\) whose Dirichlet series \(H(s) = \sum_{n \ge 1} h(n) n^{-s}\) converges absolutely for \(\operatorname{Re} s > 1/5\), such that for every complex \(s\) with \(\operatorname{Re} s > 1/2\),

\[F(s) = \frac{\zeta(2s)^2 \zeta(3s)^2}{\zeta(4s)} H(s).\]
public theorem_wanted squareful_two_pow_omega_LSeries_eq_zeta_quotient_five :
    ∃ h : ℕ → ℂ, (∀ s : ℂ, 1 / 5 < s.re → LSeriesSummable h s) ∧
      ∀ s : ℂ, 1 / 2 < s.re →
        LSeries (fun n => if Nat.IsKFull 2 n then
            (2 : ℂ) ^ ArithmeticFunction.cardDistinctFactors n else 0) s =
          riemannZeta (2 * s) ^ 2 * riemannZeta (3 * s) ^ 2 / riemannZeta (4 * s) * LSeries h s
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 2.
Uses 1
Statement uses\(k\)-full numbers

TheoremUniform fourth-moment bound for the Riemann zeta function#●ready to prove

There are real constants \(C\) and \(T_0\) such that for every real \(\sigma\) with \(1/2 \le \sigma \le 1\) and every real \(T \ge T_0\),

\[\int_1^T |\zeta(\sigma + it)|^4 \, dt \le C T (\log T)^4.\]
public theorem_wanted zeta_fourth_moment_le_uniform :
    ∃ C T₀ : ℝ, ∀ σ : ℝ, 1 / 2 ≤ σ → σ ≤ 1 → ∀ T : ℝ, T₀ ≤ T →
      ∫ t in (1 : ℝ)..T, ‖riemannZeta (σ + t * I)‖ ^ 4 ≤ C * T * Real.log T ^ 4
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 4.

TheoremVinogradov–Korobov bound for \(1/\zeta\)#●ready to prove

Write \(s = \sigma + it\) and \(g(T) = (\log T)^{2/3} (\log \log T)^{1/3}\). There are real constants \(C > 0\), \(K\) and \(T_0 \ge 3\) such that for every real \(T\) and every complex \(s\) with \(T_0 < t \le T\) and \(\sigma \ge 1 - C / g(T)\), we have \(\zeta(s) \ne 0\) and \(|1/\zeta(s)| \le K g(T)\).

public theorem_wanted inv_zeta_le_vinogradov_korobov :
    ∃ C K T₀ : ℝ, 0 < C ∧ 3 ≤ T₀ ∧ ∀ T : ℝ, ∀ s : ℂ, T₀ < s.im → s.im ≤ T →
      1 - C / (Real.log T ^ ((2 : ℝ) / 3) * Real.log (Real.log T) ^ ((1 : ℝ) / 3)) ≤ s.re →
        riemannZeta s ≠ 0 ∧ ‖(riemannZeta s)⁻¹‖ ≤
          K * Real.log T ^ ((2 : ℝ) / 3) * Real.log (Real.log T) ^ ((1 : ℝ) / 3)
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 5.

TheoremVinogradov–Korobov zero-free region#●ready to prove

There are real constants \(C > 0\) and \(t_0 \ge 3\) such that \(\zeta(\sigma + it) \ne 0\) for all real \(\sigma\) and \(t\) with \(|t| \ge t_0\) and \(\sigma \ge 1 - C / \bigl((\log |t|)^{2/3} (\log \log |t|)^{1/3}\bigr)\).

public theorem_wanted zeta_ne_zero_vinogradov_korobov :
    ∃ C t₀ : ℝ, 0 < C ∧ 3 ≤ t₀ ∧ ∀ s : ℂ, t₀ ≤ |s.im| →
      1 - C / (Real.log |s.im| ^ ((2 : ℝ) / 3) *
          Real.log (Real.log |s.im|) ^ ((1 : ℝ) / 3)) ≤ s.re →
        riemannZeta s ≠ 0
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 6.

TheoremLindelöf bound from the Riemann hypothesis#●ready to prove

Assume the Riemann hypothesis, and let \(\varepsilon > 0\) be real. Then there are real constants \(C\) and \(t_0\) such that \(|\zeta(\sigma + it)| \le C |t|^{\varepsilon}\) for all real \(\sigma\) and \(t\) with \(1/2 \le \sigma \le 1\) and \(|t| \ge t_0\).

public theorem_wanted zeta_le_rpow_of_riemannHypothesis (hRH : RiemannHypothesis)
    (ε : ℝ) (hε : 0 < ε) :
    ∃ C t₀ : ℝ, ∀ σ t : ℝ, 1 / 2 ≤ σ → σ ≤ 1 → t₀ ≤ |t| →
      ‖riemannZeta (σ + t * I)‖ ≤ C * |t| ^ ε
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 8.

TheoremBound for \(1/\zeta\) from the Riemann hypothesis#●ready to prove

Assume the Riemann hypothesis, and let \(1/2 < \sigma \le 1\) and \(\varepsilon > 0\) be real. Then there are real constants \(C\) and \(t_0\), depending on \(\sigma\) and \(\varepsilon\), such that \(|1/\zeta(\sigma + it)| \le C |t|^{\varepsilon}\) for every real \(t\) with \(|t| \ge t_0\).

public theorem_wanted inv_zeta_le_rpow_of_riemannHypothesis (hRH : RiemannHypothesis)
    (σ : ℝ) (hσ : 1 / 2 < σ) (hσ1 : σ ≤ 1) (ε : ℝ) (hε : 0 < ε) :
    ∃ C t₀ : ℝ, ∀ t : ℝ, t₀ ≤ |t| → ‖(riemannZeta (σ + t * I))⁻¹‖ ≤ C * |t| ^ ε
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Lemma 8.

TheoremAsymptotic for the sum of \(2^{\omega(n)}\) over squareful \(n\)#●ready to prove

Let \(\omega(n)\) be the number of distinct prime factors of \(n\), and let \(f(n) = 2^{\omega(n)}\) if \(n\) is squareful (\(n \ne 0\) and \(p^2 \mid n\) for every prime \(p \mid n\); so \(f(1) = 1\)) and \(f(n) = 0\) otherwise. There are real constants \(C_1\), \(C_2\), \(C_3\), \(C_4\), \(c > 0\), \(K\) and \(x_0\) such that for every real \(x \ge x_0\),

\[\Bigl|\sum_{1 \le n \le x} f(n) - \bigl(C_1 x^{1/2} \log x + C_2 x^{1/2} + C_3 x^{1/3} \log x + C_4 x^{1/3}\bigr)\Bigr| \le K x^{1/4} \exp\Bigl(c \Bigl(\frac{\log x}{\log \log x}\Bigr)^{1/3}\Bigr).\]

The paper writes the coefficient as \(c_\varepsilon\), "for some \(c > 0\)", after "let \(\varepsilon > 0\)": a constant chosen after \(\varepsilon\), not a product with an arbitrarily small factor. Its proof (equations (28)–(30)) produces, with \(T\) the height parameter of its contour, the exponent \((4 + \varepsilon) c (\log T / \log \log T)^{1/3}\) with \(\varepsilon\) small and \(c > 0\) the fixed constant of its contour shift to \(\operatorname{Re} s = 1/4 - c/g(T)\), where \(g(T) = (\log T)^{2/3} (\log \log T)^{1/3}\), chosen in terms of the Vinogradov–Korobov zero-free region and the bound for \(1/\zeta\) there (its Lemmas 5 and 6), so the coefficient stays bounded away from \(0\) and the statement is one existential \(c > 0\).

public theorem_wanted squareful_two_pow_omega_sum_asymptotic :
    ∃ C₁ C₂ C₃ C₄ c K x₀ : ℝ, 0 < c ∧ ∀ x : ℝ, x₀ ≤ x →
      |∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (if Nat.IsKFull 2 n then
            (2 : ℝ) ^ ArithmeticFunction.cardDistinctFactors n else 0) -
          (C₁ * x ^ ((1 : ℝ) / 2) * Real.log x + C₂ * x ^ ((1 : ℝ) / 2) +
            C₃ * x ^ ((1 : ℝ) / 3) * Real.log x + C₄ * x ^ ((1 : ℝ) / 3))| ≤
        K * x ^ ((1 : ℝ) / 4) *
          Real.exp (c * (Real.log x / Real.log (Real.log x)) ^ ((1 : ℝ) / 3))
Sources
  • Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan, On the Distribution of the Restricted Sequence of Integers 2^{ω(n)}, INTEGERS 24 (2024), Article A110, Theorem 1.
Uses 1
Statement uses\(k\)-full numbers
8 ready to prove

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