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Tag Archives: primes
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 numbersâ€”which is a big part of why it is so famous. Consider the … 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
Animated Sieve of Eratosthenes
Here’s something I made yesterday! (Note, I strongly suggest watching it fullscreen, in HD if you have the bandwidth for it.) Can you figure out what’s going on? The source code for the animation is here; I was inspired by … Continue reading
Posted in arithmetic, counting, pattern, pictures, primes, video
Tagged diagrams, Eratosthenes, primes, sieve, visualization
12 Comments
Fun with repunit divisors
In honor of today’s date (11/11/11), here’s a fun little problem (and some followup problems) I’ve seen posed in a few places (for example, here is a very similar problem). If I recall correctly, it was also a problem on … Continue reading
Posted in arithmetic, challenges, modular arithmetic, number theory, primes
Tagged divisors, primes, repunit
16 Comments
Prime Time in Haskell
In a recent blog post, Patrick Vennebush of Math Jokes 4 Mathy Folks noted that 2011 can be expressed as a sum of consecutive prime numbers, and challenged his readers to work out how. He also posed a couple further … Continue reading
Posted in arithmetic, number theory, primes, programming
Tagged consecutive, Haskell, primes, sum
8 Comments