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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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterEulerShift
import Mathlib.NumberTheory.EulerProduct.DirichletLSeries
namespace OAI
/-! The shifted finite character Euler factors are the actual Dirichlet
L-series factors. The logarithmic identity below uses the absolutely
convergent Euler product only in the half-plane Re(s)>1. -/
namespace TwoPointCorrelations
open Finset Filter
open scoped Classical LSeries.notation
lemma mrt_character_shifted_factor {q : ℕ} (χ : DirichletCharacter ℂ q)
(t eps : ℝ) {p : ℕ} (hp : p≠0) :
χ p*(p:ℂ)^(-(1+(eps:ℂ)-(t:ℂ)*Complex.I)) =
(characterTwist χ t p/(p:ℂ))*(Real.exp (-eps*Real.log (p:ℝ)):ℂ) := by
have hpC : (p:ℂ)≠0 := by exact_mod_cast hp
have he : -(1+(eps:ℂ)-(t:ℂ)*Complex.I) =
-1+(-(eps:ℂ)+(t:ℂ)*Complex.I) := by ring
rw [he,Complex.cpow_add _ _ hpC,Complex.cpow_neg_one,
Complex.cpow_def_of_ne_zero hpC,← Complex.natCast_log]
have hs : (Real.log (p:ℝ):ℂ)*(-(eps:ℂ)+(t:ℂ)*Complex.I) =
(-eps*Real.log (p:ℝ):ℝ)+((t*Real.log (p:ℝ):ℝ):ℂ)*Complex.I := by
push_cast
ring
rw [hs,Complex.exp_add,← Complex.ofReal_exp]
unfold characterTwist
ring
theorem mrt_character_shifted_euler_limit {q : ℕ}
(χ : DirichletCharacter ℂ q) (t : ℝ) {eps : ℝ} (heps : 0<eps) :
Tendsto (fun N : ℕ => mrtCharacterShiftedEuler χ t eps (primesUpTo N)) atTop
(nhds (L ↗χ (1+(eps:ℂ)-(t:ℂ)*Complex.I))) := by
have hs : 1<(1+(eps:ℂ)-(t:ℂ)*Complex.I).re := by
simpa using heps
have hh := χ.LSeries_eulerProduct hs
have hN : Tendsto (fun N : ℕ => N+1) atTop atTop := tendsto_add_atTop_nat 1
convert hh.comp hN using 1
funext N
change (∏ p ∈ primesUpTo N,
(1-(characterTwist χ t p/(p:ℂ))*(Real.exp (-eps*Real.log (p:ℝ)):ℂ))⁻¹) = _
change _ = ∏ p ∈ primesUpTo N,
(1-χ p*(p:ℂ)^(-(1+(eps:ℂ)-(t:ℂ)*Complex.I)))⁻¹
apply prod_congr rfl
intro p hp
rw [mrt_character_shifted_factor χ t eps (mem_filter.mp hp).2.ne_zero]
theorem mrt_character_LSeries_log_norm {q : ℕ} (χ : DirichletCharacter ℂ q)
{s : ℂ} (hs : 1<s.re) :
Summable (fun p : Nat.Primes => -Complex.log (1-χ p*(p:ℂ)^(-s))) ∧
Real.log ‖L ↗χ s‖ =
∑' p : Nat.Primes, -(Complex.log (1-χ p*(p:ℂ)^(-s))).re := by
have hsum := (summable_dirichletSummand χ hs).of_norm.clog_one_sub.neg.subtype Nat.Prime
have heq : (fun p : Nat.Primes =>
-Complex.log (1-dirichletSummandHom χ (Complex.ne_zero_of_one_lt_re hs) p)) =
(fun p : Nat.Primes => -Complex.log (1-χ p*(p:ℂ)^(-s))) := rfl
change Summable (fun p : Nat.Primes =>
-Complex.log (1-dirichletSummandHom χ (Complex.ne_zero_of_one_lt_re hs) p)) at hsum
rw [heq] at hsum
refine ⟨hsum,?_⟩
rw [← χ.LSeries_eulerProduct_exp_log hs,Complex.norm_exp,Real.log_exp,
Complex.re_tsum hsum]
congr 1
end TwoPointCorrelations
end OAI
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