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chore(IK): golf LSeries_totient_eq proof #1731

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chore(IK): golf LSeries_totient_eq proof #1731

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11 changes: 4 additions & 7 deletions PrimeNumberTheoremAnd/IwaniecKowalskiCh1.lean
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Expand Up @@ -1502,22 +1502,19 @@ lemma LSeries_totient_eq {s : ℂ} (hs : 2 < s.re) :
LSeries (↗totient) s = riemannZeta (s - 1) / riemannZeta s := by
have hs1 : 1 < s.re := by linarith
have hs2 : 1 < (s - 1).re := by
simp only [Complex.sub_re, Complex.one_re]; linarith
have hzeta_ne : riemannZeta s ≠ 0 := riemannZeta_ne_zero_of_one_lt_re hs1
have hsum_tot : LSeriesSummable (↗totientAF) s := lseriesSummable_totientAF hs
have hsum_zeta : LSeriesSummable ↗(ζ : ArithmeticFunction ℂ) s :=
LSeriesSummable_zeta_iff.mpr hs1
simp only [Complex.sub_re, Complex.one_re]
linarith
have hmul : LSeries ↗(totientAF * (ζ : ArithmeticFunction ℂ)) s =
LSeries ↗totientAF s * LSeries ↗(ζ : ArithmeticFunction ℂ) s :=
LSeries_mul' hsum_tot hsum_zeta
LSeries_mul' (lseriesSummable_totientAF hs) (LSeriesSummable_zeta_iff.mpr hs1)
have h_prod : LSeries ↗(totientAF * (ζ : ArithmeticFunction ℂ)) s = riemannZeta (s - 1) := by
rw [totientAF_mul_zeta_eq_powR1, LSeries_powR_eq 1 hs2]
have h_lzeta : LSeries ↗(ζ : ArithmeticFunction ℂ) s = riemannZeta s := by
have heq : (↗(ζ : ArithmeticFunction ℂ) : ℕ → ℂ) = ↗(ζ : ArithmeticFunction ℕ) := rfl
rw [heq]; exact LSeries_zeta_eq_riemannZeta hs1
rw [h_prod] at hmul
rw [h_lzeta] at hmul
rw [eq_div_iff hzeta_ne]
rw [eq_div_iff (riemannZeta_ne_zero_of_one_lt_re hs1)]
change LSeries (↗totientAF) s * riemannZeta s = riemannZeta (s - 1)
exact hmul.symm

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