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The binomial coefficient in the RHS enumerates the subsets of size of . The LHS does the same thing, but choosing first the largest element of , then its second-to-largest element , until choosing its smallest element .
Let denote the Möbius function and denote the floor function. Prove that:
The RHS equals
since for .
For a rational number that equals in lowest terms , let . Prove that:
First of all we note that
Moreover for we have that
Hence for we have that
The dubious function is defined as follows : and
Evaluate the sum