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KADIRI: Limit of I_2(T), equation 14 #1602

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KADIRI: Limit of I_2(T), equation 14 #1602

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37e7f40
proof(kadiri): package eq14 limit
Robby955 Jun 18, 2026
6138aa9
Drop private copies of Laplace inversion and sinc-kernel lemmas now i…
Robby955 Jun 21, 2026
4084ce9
Fix comap-negation rewrite in I_2 weight integrability
Robby955 Jun 22, 2026
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29 changes: 2 additions & 27 deletions PrimeNumberTheoremAnd/IEANTN/Kadiri.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -8,6 +8,7 @@ import PrimeNumberTheoremAnd.Mathlib.NumberTheory.LSeries.RiemannZetaHadamard
import PrimeNumberTheoremAnd.Mathlib.Analysis.SpecialFunctions.Gamma.DigammaSeries
import PrimeNumberTheoremAnd.LaplaceInversion
import PrimeNumberTheoremAnd.IEANTN.KadiriEq13
import PrimeNumberTheoremAnd.IEANTN.KadiriEq14
import Mathlib.Analysis.SpecialFunctions.Gamma.Digamma
import Mathlib.NumberTheory.LSeries.RiemannZeta

Expand Down Expand Up @@ -879,32 +880,6 @@ private theorem kadiri_thm_3_1_q1_shifted_pointwise_functional_eq
(∫ y, φ y * exp (-(-s) * (y : ℂ)) ∂volume) := by
ring

@[blueprint
"kadiri-thm-3-1-q1-I-2"
(title := "Kadiri's $I_2(T)$: the reflected Dirichlet-series piece")
(statement := /-- Kadiri's $I_2(T)$ from \cite[p.~12]{Kadiri2005}: the reflected
Dirichlet-series piece of the functional-equation rewrite of the $\sigma = -a$
integral,
$$ I_2(T) \;:=\; \frac{1}{2\pi i} \int_{-a - iT}^{-a + iT}
\frac{\zeta'}{\zeta}(1-s)\, \Phi(-s)\, ds. $$

\emph{Sign:} the $+\zeta'/\zeta(1-s)$ integrand comes from substituting the
(corrected) functional equation
$-\zeta'/\zeta(s) = -\log\pi + \zeta'/\zeta(1-s) + \tfrac{1}{2}\{\Gamma'/\Gamma(s/2)
+ \Gamma'/\Gamma((1-s)/2)\}$ (see \ref{kadiri-thm-3-1-q1-functional-eq}) into the
integrand of the $\sigma = -a$ integral and reading off the middle term. The paper
states the integrand with a leading minus, which is a typo (matching the sign typo
in the functional equation on \cite[p.~12]{Kadiri2005}). Its $T \to \infty$ limit
is given by \ref{kadiri-thm-3-1-q1-eq-14}. -/)
(latexEnv := "definition")]
noncomputable def kadiri_thm_3_1_q1_I_2 (φ : ℝ → ℂ) (a T : ℝ) : ℂ :=
let Φ : ℂ → ℂ := fun s ↦ ∫ y, φ y * exp (-s * (y : ℂ)) ∂volume
(1 / (2 * (Real.pi : ℂ))) *
∫ t in Set.Ioo (-T) T,
(deriv riemannZeta (1 - (((-a : ℝ) : ℂ) + (t : ℂ) * I)) /
riemannZeta (1 - (((-a : ℝ) : ℂ) + (t : ℂ) * I))) *
Φ (-(((-a : ℝ) : ℂ) + (t : ℂ) * I))

@[blueprint
"kadiri-thm-3-1-q1-I-3"
(title := "Kadiri's $I_3(T)$: the gamma-factor piece")
Expand Down Expand Up @@ -1450,7 +1425,7 @@ theorem kadiri_thm_3_1_q1_eq_14
Filter.Tendsto (fun T : ℝ ↦ kadiri_thm_3_1_q1_I_2 φ a T)
Filter.atTop
(nhds (-∑' n : ℕ, ((Λ n : ℂ) / (n : ℂ)) * φ (-Real.log n))) := by
sorry
exact kadiri_thm_3_1_q1_eq_14_core _hφ _hb _hφ_decay _hφ'_decay _ha _hab _ha1

/-- The digamma function commutes with complex conjugation. Mathlib's junk-value
conventions make this unconditional: `Complex.Gamma_conj` holds at every point,
Expand Down
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