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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | 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
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