| import OAI.NumberTheory.TwoPointCorrelations.Basic |
| import Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt |
| import Mathlib.NumberTheory.LSeries.Deriv |
|
|
| namespace OAI |
|
|
| /-! Exact logarithmic identities at the start of Halász's method. The |
| multiplicativity hypothesis here is complete multiplicativity on positive |
| arguments, as supplied by the corrected MRT reduction. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
|
|
| theorem halasz_logarithmic_convolution (f : ℕ → ℂ) |
| (hf : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n) (n : ℕ) : |
| f n * (Real.log (n : ℝ) : ℂ) = |
| ∑ d ∈ n.divisors, |
| (ArithmeticFunction.vonMangoldt d : ℂ) * f d * f (n / d) := by |
| by_cases hn : n = 0 |
| · subst n |
| simp |
| have hterm (d : ℕ) (hd : d ∈ n.divisors) : |
| (ArithmeticFunction.vonMangoldt d : ℂ) * f d * f (n / d) = |
| (ArithmeticFunction.vonMangoldt d : ℂ) * f n := by |
| have hdn : d ∣ n := (Nat.mem_divisors.mp hd).1 |
| have hdpos : 0 < d := Nat.pos_of_dvd_of_pos hdn (Nat.pos_of_ne_zero hn) |
| have hquot : 0 < n / d := Nat.div_pos (Nat.le_of_dvd (Nat.pos_of_ne_zero hn) hdn) hdpos |
| rw [mul_assoc, ← hf d (n / d) hdpos hquot, Nat.mul_div_cancel' hdn] |
| calc |
| _ = (∑ d ∈ n.divisors, (ArithmeticFunction.vonMangoldt d : ℂ)) * f n := by |
| rw [← Complex.ofReal_sum, ArithmeticFunction.vonMangoldt_sum] |
| ring |
| _ = _ := by rw [sum_mul]; exact sum_congr rfl hterm |>.symm |
|
|
| noncomputable def halaszDirichletPolynomial (f : ℕ → ℂ) (N : ℕ) (s : ℂ) : ℂ := |
| ∑ n ∈ Icc 1 N, LSeries.term f s n |
|
|
| theorem halaszDirichletPolynomial_hasDerivAt (f : ℕ → ℂ) (N : ℕ) (s : ℂ) : |
| HasDerivAt (halaszDirichletPolynomial f N) |
| (-∑ n ∈ Icc 1 N, (Real.log (n : ℝ) : ℂ) * LSeries.term f s n) s := by |
| change HasDerivAt (fun z => ∑ n ∈ Icc 1 N, LSeries.term f z n) _ s |
| have h := HasDerivAt.fun_sum (fun n (_ : n ∈ Icc 1 N) => LSeries.hasDerivAt_term f n s) |
| convert h using 1 |
| rw [sum_neg_distrib] |
| congr 1 |
| apply sum_congr rfl |
| intro n hn |
| have hn0 : n ≠ 0 := by have := (mem_Icc.mp hn).1; omega |
| simp [hn0, mul_div_assoc] |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
