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Tag Archives: Riemann
The Riemann zeta function and prime numbers
In a previous post I defined the famous Riemann zeta function, Today I want to give you a glimpse of what it has to do with prime numbers—which is a big part of why it is so famous. Consider the … Continue reading
The Basel problem
I wanted to follow up on something I mentioned in my previous post: I claimed that At the time I didn’t know how to prove this, but I did some quick research and today I’m going to explain it! It … Continue reading
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