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Let denote the Möbius function and denote the floor function. Prove that:
The RHS equals
since for .
Let denote Euler’s totient function. Prove that for it holds that:
where stands for the Riemann zeta function.
Well by Euler’s product we have,
Combining we get the result.
Note: It also holds that
Evaluate the limit
We are proving that the limit is . Indeed , one has:
whereas if then the above limit is . Thus:
Let denote the Möbius function. Evaluate the series
we deduce that