Weighted sums of Dirichlet convolutions

DefinitionIndicator of \(B_\kappa\)-free integers#✓fully proved

Let \(\kappa\) be a function from the natural numbers to the natural numbers (only its values \(\kappa(p)\) at primes \(p\) matter) and let \(B_\kappa\) be the set of prime powers \(p^{\kappa(p)}\). A natural number \(n\) is \(B_\kappa\)-free if \(n \ge 1\) and no \(p^{\kappa(p)}\) with \(p\) prime divides \(n\). The indicator \(q_{B_\kappa}\) is the integer-valued arithmetic function with \(q_{B_\kappa}(n) = 1\) if \(n\) is \(B_\kappa\)-free and \(q_{B_\kappa}(n) = 0\) otherwise.

The paper indexes the exponents by the position of the prime (\(b_i = p_i^{\kappa_i}\)); here \(\kappa\) is indexed by the prime itself. For constant \(\kappa = k\) this is the indicator \(q_k\) of \(k\)-free integers.

public noncomputable def bFreeIndicator (κ : ℕ → ℕ) : ArithmeticFunction ℤ :=
  ⟨fun n => if n ≠ 0 ∧ ∀ p : ℕ, p.Prime → ¬ p ^ κ p ∣ n then 1 else 0, by simp⟩
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Equations (2), (11), (12).
Used by 8
Used byThe \(B_\kappa\)-free indicator twisted by the principal character, convolved with \(\mu\) · Asymptotic for the divisor function summed over \(B_\kappa\)-free integers · Asymptotic for the divisor function summed over \(B_\kappa\)-free integers in a progression · Values of the \(B_\kappa\)-free indicator · The \(B_\kappa\)-free indicator convolved with the Möbius function · The integer \(1\) is \(B_\kappa\)-free · The prime power \(p^{\kappa(p)}\) is not \(B_\kappa\)-free · Asymptotic for the divisor function summed over \(k\)-free integers

TheoremValues of the \(B_\kappa\)-free indicator#✓fully proved

Let \(\kappa\) be a function from the natural numbers to the natural numbers. For every natural number \(n\), the indicator \(q_{B_\kappa}\) of \(B_\kappa\)-free integers satisfies \(q_{B_\kappa}(n) = 1\) if \(n \ne 0\) and no \(p^{\kappa(p)}\) with \(p\) prime divides \(n\), and \(q_{B_\kappa}(n) = 0\) otherwise.

public theorem bFreeIndicator_apply (κ : ℕ → ℕ) (n : ℕ) :
    bFreeIndicator κ n = if n ≠ 0 ∧ ∀ p : ℕ, p.Prime → ¬ p ^ κ p ∣ n then 1 else 0
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Equations (11), (12).
Uses 1 · Used by 2
Statement usesIndicator of \(B_\kappa\)-free integers
Used byThe integer \(1\) is \(B_\kappa\)-free · The prime power \(p^{\kappa(p)}\) is not \(B_\kappa\)-free

TheoremThe integer \(1\) is \(B_\kappa\)-free#✓fully proved

Let \(\kappa\) be a function from the natural numbers to the natural numbers with \(\kappa(p) \ge 1\) for every prime \(p\). Then the indicator \(q_{B_\kappa}\) of \(B_\kappa\)-free integers satisfies \(q_{B_\kappa}(1) = 1\).

public theorem bFreeIndicator_one {κ : ℕ → ℕ} (hκ : ∀ p : ℕ, p.Prime → 1 ≤ κ p) :
    bFreeIndicator κ 1 = 1
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Equation (12).
Uses 2
Statement usesIndicator of \(B_\kappa\)-free integers
Proof usesValues of the \(B_\kappa\)-free indicator

TheoremThe prime power \(p^{\kappa(p)}\) is not \(B_\kappa\)-free#✓fully proved

Let \(\kappa\) be a function from the natural numbers to the natural numbers and let \(p\) be a prime. Then the indicator \(q_{B_\kappa}\) of \(B_\kappa\)-free integers satisfies \(q_{B_\kappa}(p^{\kappa(p)}) = 0\).

No positivity of \(\kappa(p)\) is assumed; for \(\kappa(p) = 0\) the statement is \(q_{B_\kappa}(1) = 0\).

public theorem bFreeIndicator_prime_pow {κ : ℕ → ℕ} {p : ℕ} (hp : p.Prime) :
    bFreeIndicator κ (p ^ κ p) = 0
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Equation (12).
Uses 2
Statement usesIndicator of \(B_\kappa\)-free integers
Proof usesValues of the \(B_\kappa\)-free indicator

TheoremProduct formula for homogeneous sums of powers#●ready to prove

Let \(R\) be a commutative ring, let \(a, b \in R\) and let \(\tau, \xi\) be natural numbers. Then

\[ \sum_{j=0}^{\tau+\xi} a^j b^{\tau+\xi-j} = \left(\sum_{j=0}^{\tau} a^j b^{\tau-j}\right) \left(\sum_{i=0}^{\xi} a^i b^{\xi-i}\right) - ab \left(\sum_{j=0}^{\tau-1} a^j b^{\tau-1-j}\right) \left(\sum_{i=0}^{\xi-1} a^i b^{\xi-1-i}\right), \]

where a sum with upper limit \(-1\) is zero.

In the source, \(a\) and \(b\) are the values \(u(n)\) and \(v(n)\) of complex arithmetic functions.

public theorem_wanted sum_pow_mul_pow_add_eq {R : Type*} [CommRing R] (a b : R) (τ ξ : ℕ) :
    ∑ j ∈ Finset.range (τ + ξ + 1), a ^ j * b ^ (τ + ξ - j) =
      (∑ j ∈ Finset.range (τ + 1), a ^ j * b ^ (τ - j)) *
          (∑ i ∈ Finset.range (ξ + 1), a ^ i * b ^ (ξ - i)) -
        a * b * (∑ j ∈ Finset.range τ, a ^ j * b ^ (τ - 1 - j)) *
          (∑ i ∈ Finset.range ξ, a ^ i * b ^ (ξ - 1 - i))
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Lemma 1, Equation (16).

TheoremGcd expansion of a convolution of completely multiplicative functions at a product#●ready to prove

Let \(u\) and \(v\) be complex-valued completely multiplicative arithmetic functions, meaning \(u(mn) = u(m) u(n)\) and \(v(mn) = v(m) v(n)\) for all natural numbers \(m\) and \(n\). Let \(u * v\) be their Dirichlet convolution and \(\mu\) the Möbius function. Then for all natural numbers \(r\) and \(s\),

\[(u * v)(rs) = \sum_{m \mid \gcd(r, s)} \mu(m) u(m) v(m) (u * v)(r/m) (u * v)(s/m),\]

the sum running over the positive divisors \(m\) of \(\gcd(r, s)\) (it is empty when \(r = s = 0\)).

The paper takes \(r, s\) positive; the identity also holds (both sides vanish) when \(r\) or \(s\) is \(0\), so no positivity hypothesis is imposed.

public theorem_wanted dirichletConvolution_mul_eq_sum_gcd {u v : ArithmeticFunction ℂ}
    (hu : u.IsCompletelyMultiplicative) (hv : v.IsCompletelyMultiplicative) (r s : ℕ) :
    (u * v) (r * s) = ∑ m ∈ (Nat.gcd r s).divisors,
      (μ m : ℂ) * u m * v m * (u * v) (r / m) * (u * v) (s / m)
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Lemma 2, Equation (17).
Uses 1
Statement usesCompletely multiplicative arithmetic function

TheoremSums of a Dirichlet convolution over multiples of \(r\)#●ready to prove

Let \(u\) and \(v\) be completely multiplicative arithmetic functions with complex values, let \(u * v\) be their Dirichlet convolution, let \(\mu\) be the Möbius function, and for real \(y\) put \(D(y) = \sum_{1 \le j \le y} (u * v)(j)\). Then for every natural number \(r\) and every real \(x\),

\[ \sum_{1 \le j \le x/r} (u * v)(rj) = \sum_{d \mid r} u(d)\, v(r/d) \sum_{m \mid d} \mu(m)\, v(m)\, D\left(\frac{x}{mr}\right), \]

where \(d\) and \(m\) run over positive divisors and \(x/0 = 0\), so both sides vanish for \(r = 0\).

Both sides also vanish for \(x < 1\); the source assumes \(r \ge 1\) and \(x \ge 1\).

public theorem_wanted sum_dirichletConvolution_mul_eq {u v : ArithmeticFunction ℂ}
    (hu : u.IsCompletelyMultiplicative) (hv : v.IsCompletelyMultiplicative) (r : ℕ) (x : ℝ) :
    ∑ j ∈ Finset.Icc 1 ⌊x / r⌋₊, (u * v) (r * j) =
      ∑ d ∈ r.divisors, u d * v (r / d) * ∑ m ∈ d.divisors, (μ m : ℂ) * v m *
        ∑ j ∈ Finset.Icc 1 ⌊x / (m * r)⌋₊, (u * v) j
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Lemma 3, Equation (20).
Uses 1
Statement usesCompletely multiplicative arithmetic function

TheoremMöbius expansion of weighted sums of a Dirichlet convolution#●ready to prove

Let \(w\), \(u\) and \(v\) be arithmetic functions with complex values, let \(*\) denote Dirichlet convolution, let \(\mu\) be the Möbius function and put \(g = w * \mu\). Then for every real \(x\),

\[ \sum_{1 \le j \le x} w(j)\, (u * v)(j) = \sum_{1 \le r \le x} g(r) \sum_{1 \le j \le x/r} (u * v)(rj), \]

where \(j\) and \(r\) run over positive integers.

Both sides vanish for \(x < 1\); the source assumes \(x \ge 1\).

public theorem_wanted sum_mul_dirichletConvolution_eq_sum_moebius (w u v : ArithmeticFunction ℂ)
    (x : ℝ) :
    ∑ j ∈ Finset.Icc 1 ⌊x⌋₊, w j * (u * v) j =
      ∑ r ∈ Finset.Icc 1 ⌊x⌋₊, (w * (μ : ArithmeticFunction ℂ)) r *
        ∑ j ∈ Finset.Icc 1 ⌊x / r⌋₊, (u * v) (r * j)
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Lemma 4.

TheoremWeighted summation of Dirichlet convolutions#●ready to prove

Let \(u\) and \(v\) be completely multiplicative arithmetic functions with complex values, let \(w\) be any arithmetic function with complex values, let \(*\) denote Dirichlet convolution, let \(\mu\) be the Möbius function, put \(g = w * \mu\), and for real \(y\) put \(D(y) = \sum_{1 \le j \le y} (u * v)(j)\). Then for every real \(x\),

\[ \sum_{1 \le j \le x} w(j)\, (u * v)(j) = \sum_{1 \le r \le x} g(r) \sum_{d \mid r} u(d)\, v(r/d) \sum_{m \mid d} \mu(m)\, v(m)\, D\left(\frac{x}{mr}\right), \]

where \(r\), \(d\) and \(m\) run over positive integers.

Both sides vanish for \(x < 1\); the source assumes \(x \ge 1\).

public theorem_wanted weighted_sum_dirichletConvolution_eq {u v : ArithmeticFunction ℂ}
    (hu : u.IsCompletelyMultiplicative) (hv : v.IsCompletelyMultiplicative)
    (w : ArithmeticFunction ℂ) (x : ℝ) :
    ∑ j ∈ Finset.Icc 1 ⌊x⌋₊, w j * (u * v) j =
      ∑ r ∈ Finset.Icc 1 ⌊x⌋₊, (w * (μ : ArithmeticFunction ℂ)) r *
        ∑ d ∈ r.divisors, u d * v (r / d) * ∑ m ∈ d.divisors, (μ m : ℂ) * v m *
          ∑ j ∈ Finset.Icc 1 ⌊x / (m * r)⌋₊, (u * v) j
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Theorem 1, Equation (8).
Uses 1
Statement usesCompletely multiplicative arithmetic function

TheoremThe \(B_\kappa\)-free indicator convolved with the Möbius function#●ready to prove

Let \(\kappa\) be a function from the natural numbers to the natural numbers with \(\kappa(p) \ge 1\) for every prime \(p\), let \(q_{B_\kappa}\) be the indicator of \(B_\kappa\)-free integers, let \(\mu\) be the Möbius function and let \(*\) denote Dirichlet convolution. Let \(n \ge 1\) be a natural number with prime factorization \(n = \prod_p p^{e_p}\). If \(\kappa(p)\) divides \(e_p\) for every prime \(p\), then \((q_{B_\kappa} * \mu)(n) = \mu(n^{1/\kappa})\), where \(n^{1/\kappa} = \prod_p p^{e_p/\kappa(p)}\); otherwise \((q_{B_\kappa} * \mu)(n) = 0\).

The source indexes the exponents by the position of the prime (\(\kappa_i\) for the \(i\)-th prime \(p_i\)) and requires \(b_i = p_i^{\kappa_i} > 1\), which is the hypothesis \(\kappa(p) \ge 1\).

public theorem_wanted bFreeIndicator_mul_moebius_apply {κ : ℕ → ℕ}
    (hκ : ∀ p : ℕ, p.Prime → 1 ≤ κ p) {n : ℕ} (hn : 1 ≤ n) :
    (bFreeIndicator κ * μ) n =
      if ∀ p : ℕ, κ p ∣ n.factorization p then
        μ (n.factorization.prod fun p e => p ^ (e / κ p))
      else 0
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Lemma 5, Equation (22).
Uses 1
Statement usesIndicator of \(B_\kappa\)-free integers

TheoremThe \(B_\kappa\)-free indicator twisted by the principal character, convolved with \(\mu\)#●ready to prove

Let \(\kappa\) be a function from the natural numbers to the natural numbers with \(\kappa(p) \ge 1\) for every prime \(p\), let \(\xi\) be a natural number, let \(q_{B_\kappa}\) be the indicator of \(B_\kappa\)-free integers and let \(\mu\) be the Möbius function. Put \(w(d) = q_{B_\kappa}(d)\) if \(\gcd(d, \xi) = 1\) and \(w(d) = 0\) otherwise, so that \(w = q_{B_\kappa} \chi_\xi\) with \(\chi_\xi\) the principal character modulo \(\xi\), and let \((w * \mu)(n) = \sum_{d \mid n} w(d) \mu(n/d)\), the sum running over the positive divisors \(d\) of \(n\) (it is empty for \(n = 0\)). For \(r \ge 1\) with prime factorization \(r = \prod_p p^{e_p}\) write \(r^\kappa = \prod_p p^{\kappa(p) e_p}\), and set \(0^\kappa = 1\). Then for every natural number \(n\) with \((w * \mu)(n) \ne 0\) there are natural numbers \(\ell\) and \(r\) with \(\ell \mid \xi\), \(\gcd(r, \xi) = 1\) and \(n = \ell r^\kappa\). Moreover, for all natural numbers \(\ell\) and \(r\) with \(\ell \mid \xi\), \(\gcd(r, \xi) = 1\) and \(r \ge 1\), \((w * \mu)(\ell r^\kappa) = \mu(\ell) \mu(r)\).

The hypothesis \(r \ge 1\) in the evaluation is needed: for \(r = 0\) the formula is false. No positivity of \(\xi\) is assumed.

public theorem_wanted bFreeCoprimeIndicator_mul_moebius {κ : ℕ → ℕ}
    (hκ : ∀ p : ℕ, p.Prime → 1 ≤ κ p) (ξ : ℕ) :
    (∀ n : ℕ, ∑ d ∈ n.divisors, (if d.Coprime ξ then bFreeIndicator κ d else 0) * μ (n / d) ≠ 0 →
        ∃ ℓ r : ℕ, ℓ ∣ ξ ∧ r.Coprime ξ ∧ n = ℓ * r.factorization.prod fun p e => p ^ (κ p * e)) ∧
      ∀ ℓ r : ℕ, ℓ ∣ ξ → r.Coprime ξ → 0 < r →
        ∑ d ∈ (ℓ * r.factorization.prod fun p e => p ^ (κ p * e)).divisors,
            (if d.Coprime ξ then bFreeIndicator κ d else 0) *
              μ ((ℓ * r.factorization.prod fun p e => p ^ (κ p * e)) / d) = μ ℓ * μ r
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Lemma 6, Equation (23).
Uses 1
Statement usesIndicator of \(B_\kappa\)-free integers

TheoremAsymptotic for the divisor function summed over \(k\)-free integers#●ready to prove

Let \(k \ge 2\) be a natural number, let \(q_k\) be the indicator of \(k\)-free integers (the indicator of \(B_\kappa\)-free integers for the constant function \(\kappa = k\)), let \(\tau(n)\) be the number of positive divisors of \(n\), let \(\gamma\) be the Euler–Mascheroni constant, let \(\Delta(x) = \sum_{1 \le n \le x} \tau(n) - x \log x - (2\gamma - 1) x\) be the Dirichlet divisor error term, and put

\[c_k = \prod_p \Bigl(1 - \frac{k + 1}{p^k} + \frac{k}{p^{k+1}}\Bigr),\]

the product running over all primes \(p\). Then there is a real constant \(\beta_k\) such that \(E(x) = \sum_{1 \le n \le x} q_k(n) \tau(n) - c_k x \log x - \beta_k x\) satisfies the following as \(x \to \infty\). If \(k \le 3\), then \(E(x) = O(x^{1/k + \varepsilon})\) for every \(\varepsilon > 0\). If \(k \ge 4\), then for every real \(\eta \ge 1/4\) with \(\Delta(x) = O(x^\eta)\), \(E(x) = O(x^{\eta + \varepsilon})\) for every \(\varepsilon > 0\).

public theorem_wanted kFree_divisor_sum_asymptotic {k : ℕ} (hk : 2 ≤ k) :
    ∃ β : ℝ,
      let err : ℝ → ℝ := fun x =>
        (∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (bFreeIndicator (fun _ => k) n : ℝ) * n.divisors.card) -
          (∏' p : Nat.Primes, (1 - (k + 1) / (p : ℝ) ^ k + k / (p : ℝ) ^ (k + 1))) *
            x * Real.log x - β * x
      (k ≤ 3 → ∀ ε : ℝ, 0 < ε → err =O[atTop] fun x => x ^ ((1 : ℝ) / k + ε)) ∧
        (4 ≤ k → ∀ η : ℝ, 1 / 4 ≤ η →
          ((fun x : ℝ => (∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (n.divisors.card : ℝ)) - x * Real.log x -
              (2 * Real.eulerMascheroniConstant - 1) * x) =O[atTop] fun x => x ^ η) →
            ∀ ε : ℝ, 0 < ε → err =O[atTop] fun x => x ^ (η + ε))
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Corollary 1.
Uses 1
Statement usesIndicator of \(B_\kappa\)-free integers

TheoremAsymptotic for the divisor function summed over \(B_\kappa\)-free integers#●ready to prove

Let \(\kappa\) be a function from the natural numbers to the natural numbers and let \(K \ge 2\) be the minimum of \(\kappa(p)\) over the primes \(p\): \(\kappa(p) \ge K\) for every prime \(p\), with equality for some prime. Let \(q_{B_\kappa}\) be the indicator of \(B_\kappa\)-free integers, \(\tau(n)\) the number of positive divisors of \(n\), \(\gamma\) the Euler–Mascheroni constant and \(\Delta(x) = \sum_{1 \le n \le x} \tau(n) - x \log x - (2\gamma - 1) x\) the Dirichlet divisor error term, and put

\[C_\kappa = \prod_p \Bigl(1 - \frac{\kappa(p) + 1}{p^{\kappa(p)}} + \frac{\kappa(p)}{p^{\kappa(p)+1}}\Bigr),\]

the product running over all primes \(p\). Then there is a real constant \(\beta_\kappa\) such that \(E(x) = \sum_{1 \le n \le x} q_{B_\kappa}(n) \tau(n) - C_\kappa x \log x - \beta_\kappa x\) satisfies the following as \(x \to \infty\). If \(K \le 3\), then \(E(x) = O(x^{1/K + \varepsilon})\) for every \(\varepsilon > 0\). If \(K \ge 4\), then for every real \(\eta \ge 1/4\) with \(\Delta(x) = O(x^\eta)\), \(E(x) = O(x^{\eta + \varepsilon})\) for every \(\varepsilon > 0\).

public theorem_wanted bFree_divisor_sum_asymptotic {κ : ℕ → ℕ} {K : ℕ} (hK : 2 ≤ K)
    (hmin : ∀ p : ℕ, p.Prime → K ≤ κ p) (hatt : ∃ p : ℕ, p.Prime ∧ κ p = K) :
    ∃ β : ℝ,
      let err : ℝ → ℝ := fun x =>
        (∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (bFreeIndicator κ n : ℝ) * n.divisors.card) -
          (∏' p : Nat.Primes, (1 - (κ p + 1) / (p : ℝ) ^ κ p + κ p / (p : ℝ) ^ (κ p + 1))) *
            x * Real.log x - β * x
      (K ≤ 3 → ∀ ε : ℝ, 0 < ε → err =O[atTop] fun x => x ^ ((1 : ℝ) / K + ε)) ∧
        (4 ≤ K → ∀ η : ℝ, 1 / 4 ≤ η →
          ((fun x : ℝ => (∑ n ∈ Finset.Icc 1 ⌊x⌋₊, (n.divisors.card : ℝ)) - x * Real.log x -
              (2 * Real.eulerMascheroniConstant - 1) * x) =O[atTop] fun x => x ^ η) →
            ∀ ε : ℝ, 0 < ε → err =O[atTop] fun x => x ^ (η + ε))
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Corollary 2.
Uses 1
Statement usesIndicator of \(B_\kappa\)-free integers

TheoremAsymptotic for the divisor function summed over \(B_\kappa\)-free integers in a progression#●ready to prove

Let \(\kappa\) be a function from the natural numbers to the natural numbers and let \(K\) be the minimum of \(\kappa(p)\) over the primes \(p\): \(\kappa(p) \ge K\) for every prime \(p\), with equality for some prime. Assume \(K \ge 2\) and \(K \ne 3\). Let \(a\) and \(\xi\) be natural numbers with \(a < \xi\) and \(\gcd(a, \xi) = 1\). Let \(q_{B_\kappa}\) be the indicator of \(B_\kappa\)-free integers, \(\tau(n)\) the number of positive divisors of \(n\) and \(\varphi\) Euler's totient function, and put

\[C_{\kappa,\xi} = \prod_{p \nmid \xi} \Bigl(1 - \frac{\kappa(p) + 1}{p^{\kappa(p)}} + \frac{\kappa(p)}{p^{\kappa(p)+1}}\Bigr),\]

the product running over the primes \(p\) that do not divide \(\xi\). Let \(\eta = 1/2\) if \(K = 2\) and \(\eta = 1/3\) if \(K \ge 4\). Then there is a real constant \(\tilde c\) such that for every \(\varepsilon > 0\), as \(x \to \infty\),

\[\sum_{\substack{1 \le n \le x \\ n \equiv a \pmod{\xi}}} q_{B_\kappa}(n) \tau(n) = \frac{\varphi(\xi)}{\xi^2} C_{\kappa,\xi} \, x \log x + \tilde c x + O(x^{\eta + \varepsilon}).\]

The case \(K = 3\) is not covered by the paper's statement and is excluded.

public theorem_wanted bFree_divisor_sum_progression_asymptotic {κ : ℕ → ℕ} {K : ℕ} (hK : 2 ≤ K)
    (hK3 : K ≠ 3) (hmin : ∀ p : ℕ, p.Prime → K ≤ κ p) (hatt : ∃ p : ℕ, p.Prime ∧ κ p = K)
    {a ξ : ℕ} (haξ : a < ξ) (hcop : a.Coprime ξ) :
    ∃ c : ℝ, ∀ ε : ℝ, 0 < ε →
      (fun x : ℝ =>
        (∑ n ∈ (Finset.Icc 1 ⌊x⌋₊).filter (fun n => n % ξ = a),
            (bFreeIndicator κ n : ℝ) * n.divisors.card) -
          (Nat.totient ξ : ℝ) / (ξ : ℝ) ^ 2 *
            (∏' p : Nat.Primes, if (p : ℕ).Coprime ξ then
              (1 - (κ p + 1) / (p : ℝ) ^ κ p + κ p / (p : ℝ) ^ (κ p + 1)) else 1) *
            x * Real.log x - c * x) =O[atTop]
        fun x => x ^ ((if K = 2 then (1 : ℝ) / 2 else 1 / 3) + ε)
Sources
  • André Pierro de Camargo, Weighted Summation of Dirichlet Convolutions and Applications to Some Divisor Problems over k-Free and B-Free Integers, INTEGERS 24 (2024), Article A75, Corollary 3, Equation (15).
Uses 1
Statement usesIndicator of \(B_\kappa\)-free integers
4 fully proved10 ready to prove

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