Tag Archives: convolution

The Riemann zeta function

Posted on February 2, 2017 by Brent

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 , , , , , , , | 2 Comments

Dirichlet generating functions

Posted on January 30, 2017 by Brent

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 , , , , , , , | 5 Comments

More fun with Dirichlet convolution

Posted on January 5, 2017 by Brent

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 , , , , , , | 1 Comment

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