Journal of Integer Sequences, Vol. 25 (2022), Article 22.7.3

On Landau's Inequality for the Prime Counting Function

Horst Alzer
Morsbacher Straß 10
51545 Waldbröl

Man Kam Kwong
Department of Applied Mathematics
The Hong Kong Polytechnic University
Hong Kong


We prove that the inequality 2π(n) – π(2n) ≥ 2ω(n) is valid for all n ≥ 71. Here, π(n) denotes the prime counting function and ω(n) denotes the number of distinct prime factors of n. Our inequality refines a recently published result by Zhang.

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(Concerned with sequences A000720 A001221 A060208.)

Received June 28 2022; revised versions received September 8 2022; September 10 2022. Published in Journal of Integer Sequences, September 11 2022.

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