Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Tmp

BrunTitchmarsh.tmp

analytic number theory prime numbers asymptotics brun titchmarsh

Plain-language statement

An explicit upper bound for the number of primes below $N$ : $|\{p<N:p\text{ is prime}\}|\le \frac{4N}{\log N}+6\sqrt N\left(1+\frac12\log N\right)^3.$ The formal statement also covers the small values of $N$ using Lean's totalized real operations.

Exact Lean statement

theorem tmp (N : ℕ) :
    ((Finset.range N).filter Nat.Prime).card ≤
      4 * (N / Real.log N) + 6 * (N ^ (1/2 : ℝ) * (1 + 1/2 * Real.log N) ^ 3)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem tmp (N : ℕ) :    ((Finset.range N).filter Nat.Prime).card ≤      4 * (N / Real.log N) + 6 * (N ^ (1/2 : ℝ) * (1 + 1/2 * Real.log N) ^ 3) := by  trans ↑((Finset.range (N+1)).filter Nat.Prime).card  · norm_cast    refine Finset.card_le_card fun n ↦ ?_    simp only [Finset.mem_filter, Finset.mem_range, and_imp]    exact fun hnN hp ↦ ⟨by omega, hp⟩  rw [← primesBetween_one]  by_cases hN : N = 0  · simp [hN, primesBetween]  by_cases hN : N = 1  · simp (config := {decide:=true}) [hN, primesBetween]  have h : primesBetween 1 (1 + N) ≤      2 * (N / Real.log (N ^ (1/2 : ℝ))) +        6 * (N ^ (1 / 2 : ℝ) * (1 + Real.log (N ^ (1/2 : ℝ))) ^ 3) := by    convert (primesBetween_le 1 N (N ^ (1/2 : ℝ)) (by norm_num) (by norm_cast; omega)      (Real.one_lt_rpow (by norm_cast; omega) (by norm_num))) using 1    ring  calc    _ ≤ (primesBetween 1 (1+N):ℝ) := by      norm_cast      apply primesBetween_mono_right      norm_cast      omega    _ ≤ _ := by      rw [Real.log_rpow (by norm_num; omega)] at h      convert h using 2      ring

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/BrunTitchmarsh.lean:274-302

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Admissible bound mono

admissible_bound.mono

Plain-language statement

For positive parameters $A,B,C,R$ , the classical error-bound function $A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ is nonincreasing once $x\ge \exp\!\left(R(2B/C)^2\right)$ .

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Analytic On div Removable zero

AnalyticOn_divRemovable_zero

Plain-language statement

Let $f$ be analytic on an open set $s$ containing $0$ , and suppose $f(0)=0$ . Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$ . Then the apparent singularity at $0$ is removable and $g$ is analytic throughout $s$ .

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Analytic On div Removable zero closed Ball

AnalyticOn_divRemovable_zero_closedBall

Plain-language statement

Suppose $R>0$ and $f$ is analytic on the closed disc $|z|\le R$ with $f(0)=0$ . Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$ . Then $g$ is analytic on the entire closed disc, including at the removed singularity.

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record