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Nightmare NT comes again

Source: 2024 Chinese TST P3

March 6, 2024
number theory2024 CTST

Problem Statement

Given positive integer M.M. For any n∈N+,n\in\mathbb N_+, let h(n)h(n) be the number of elements in [n][n] that are coprime to M.M. Define β:=h(M)M.\beta :=\frac {h(M)}M. Proof: there are at least M3\frac M3 elements nn in [M],[M], satisfy ∣h(n)−βn∣≤β⋅2ω(M)−3+1.\left| h(n)-\beta n\right|\le\sqrt{\beta\cdot 2^{\omega(M)-3}}+1. Here [n]:={1,2,…,n}[n]:=\{1,2,\ldots ,n\} for all positive integer n.n. Proposed by Bin Wang
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