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Tag Archives: mu
The Riemann zeta function
Recall from my previous post that given a function , we define , the Dirichlet generating function of , by We also proved that : the product of Dirichlet generating functions is the Dirichlet generating function of the Dirichlet convolution. … Continue reading
Posted in number theory
Tagged convolution, Dirichlet, inversion, moebius, mu, primes, Riemann, zeta
2 Comments
Dirichlet generating functions
Suppose is a function defined for positive integers . Then we can define an infinite series as follows: (This might look a bit strange, but bear with me!) For example, suppose for all . Then (Note that in this case, … Continue reading
Posted in number theory
Tagged convolution, Dirichlet, inversion, moebius, mu, primes, Riemann, zeta
5 Comments
More fun with Dirichlet convolution
I’m back after a bit of a hiatus for the holidays! Last time we saw how the principle of Möbius inversion arises from considering the function from the point of view of Dirichlet convolution. Put simply, the Möbius function is … Continue reading
Posted in number theory
Tagged arithmetic, convolution, Dirichlet, divisors, inversion, moebius, mu
1 Comment
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
3 Comments
The Möbius function proof, part 1
In my last post, I introduced the Möbius function , which is defined in terms of the prime factorization of : if has any repeated prime factors, that is, if is divisible by a perfect square. Otherwise, if has distinct … Continue reading
The Möbius function
Time to pull back the curtain a bit! My recent series of posts on complex roots of unity may seem somewhat random and unmotivated so far, but the fact is that I definitely have a destination in mind—we are slowly … Continue reading