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Lower estimate on Sum gcd(i,j)

Source: Kürschák 2003, problem 3

July 8, 2014
inequalitiesnumber theoryAnalytic Number Theoryexpected valueProbabilistic MethodSummationphi function

Problem Statement

Prove that the following inequality holds with the exception of finitely many positive integers nn:
∑i=1n∑j=1ngcd(i,j)>4n2.\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.
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