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Monthly Archives: December 2016
Möbius inversion
In my last post we saw that , that is, the Möbius function is the inverse of with respect to Dirichlet convolution. This directly leads to an interesting principle called Möbius inversion. Möbius inversion. Suppose is defined for as the … Continue reading
Dirichlet convolution and the Möbius function
Recall from last time that the Dirichlet convolution of two functions and is written and defined by: where the sum is taken over all possible factorizations of into a product of positive integers. Last time we saw that is commutative … Continue reading
Dirichlet convolution
Let and be two functions defined on the positive integers. Then the Dirichlet convolution of and , written , is another function on the positive integers, defined as follows: The sum is taken over all possible factorizations of into a … Continue reading
The Möbius function proof, part 2 (the subset parity lemma)
Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading
Posted in arithmetic, combinatorics, complex numbers, primes, proof
Tagged circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity
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