Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Γ₃ lower bound

γ₃_lower_bound

analytic number theory prime numbers asymptotics euler mascheroni bounds

Plain-language statement

Let $\gamma_3(n)=H_n-\log n-\frac{1}{2n}+\frac{1}{12n^2}-\frac{1}{120n^4}.$ For every integer $n\ge1$ , this corrected harmonic approximation is a strict lower bound for the Euler–Mascheroni constant: $\gamma_3(n)<\gamma.$

Exact Lean statement

lemma γ₃_lower_bound (n : ℕ) (hn : 1 ≤ n) :
    γ₃ n < Real.eulerMascheroniConstant

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma γ₃_lower_bound (n : ℕ) (hn : 1 ≤ n) :    γ₃ n < Real.eulerMascheroniConstant := by  have h_strict_mono : StrictMono (fun n : ℕ => γ₃ (n + 1)) :=    strictMono_nat_of_lt_succ (fun k => γ₃_increasing (k + 1) (by omega))  have h_tendsto : Filter.Tendsto (fun n : ℕ => γ₃ (n + 1))      Filter.atTop (nhds eulerMascheroniConstant) :=    γ₃_tendsto.comp (Filter.tendsto_add_atTop_nat 1)  have h_lt : ∀ k : ℕ, γ₃ (k + 1) < eulerMascheroniConstant := fun k =>    lt_of_lt_of_le      (h_strict_mono <| Nat.lt_succ_self _)      (ge_of_tendsto h_tendsto <| Filter.eventually_atTop.mpr        ⟨k + 1, fun m hm => h_strict_mono.monotone hm⟩)  cases n with  | zero => omega  | succ m => exact h_lt m

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/EulerMascheroniBounds.lean:471-485

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View proof record