| 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 |
|
|
