Chebyshev's theorem

Proper noun

The theorem that the prime counting function is of the same order of magnitude as x / ln x, i.e., for the prime counting function π, there are positive constants c and C such that:

Japanese Meaning

チェビシェフの定理とは、素数計数関数 π(x) が x/ln x と同じオーダーで増加するという定理です。すなわち、十分大きな x に対して、正の定数 c および C が存在し、c·(x/ln x) ≤ π(x) ≤ C·(x/ln x) となるという主張を示しています。

What is this buttons?

Chebyshev's theorem was a breakthrough result that showed the prime counting function π(x) is of the same order of magnitude as x and provided the first effective bounds toward the prime number theorem.

Google Translation / DeepL Translation

素数計数関数が x と同じオーダーであるという定理は画期的な成果で、π(x) が x と同じオーダーで増加することを示し、素数定理に向けた最初の有効な評価を与えました。

Details

Sense（1）

The theorem that the prime counting function is of the same order of magnitude as x / ln x, i.e., for the prime counting function π, there are positive constants c and C such that:

∀x∈ℕ:(cx)/(ln x)<=𝜋(x)<=(Cx)/(ln x)

Google Translation / DeepL Translation

Details

Sense（2）

Bertrand's postulate, as proven by Chebyshev.

Sense（3）

Chebyshev's inequality.

Details

Chebyshev's theorem

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