Monthly Archives: December 2016

Möbius inversion

Posted on December 14, 2016 by Brent

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 →

Posted in combinatorics, proof | Tagged , , , , | 3 Comments

Dirichlet convolution and the Möbius function

Posted on December 11, 2016 by Brent

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 →

Posted in combinatorics, proof | Tagged , , , | 1 Comment

Dirichlet convolution

Posted on December 6, 2016 by Brent

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 →

Posted in combinatorics, proof | Tagged , , , | 8 Comments

The Möbius function proof, part 2 (the subset parity lemma)

Posted on December 3, 2016 by Brent

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 , , , , , , , , , | 3 Comments