Number Theory

I'm in the process of putting together much clearer, much cleaner expositions of my own personal explorations of number theory. So this is just a historical collection of just a few of my earlier explorations.

This video represents some of my more recent work on math exposition. The actual video is somewhat incomplete, as I had to push it out for a deadline. In particular, it's missing a bunch of animations I still have unpublished about Poisson summation, which I think are really striking. And the exposition itself is, to be blunt, a mess. But I think a lot of the animations are compelling.

• Prime Counting in O(n^2/3 log n) time and O(n^1/3 log n) space with Linnik's identity

  (PDF, 2011-23-2011) This is a write up of an algorithm I've put together for counting the number of primes less than some number n, relying on Linnik's identity and properties of the divisor summatory functions. This paper includes both a detailed description of the algorithm as well as a function C implementation that demonstrates its running time properties.
  

• A Few Novel, Quick Prime Number Counting Techniques

  (PDF, 2012-12-5) This paper presents several prime counting approaches all stemming from Linnik's identity. It includes both Mathematica and C code.
  



• Extending My Prime Counting Algorithm to Computing the Sums of Primes

  (PDF, 2011-11-23) My prime counting algorithm can actually be generalized to compute any non-negative integer prime sums in O(n^2/3 log n) time and O(n^1/3 log n) space. This is an informal and rough write-up of how such an algorithm works. A rough implementation of this idea (with some shaky precision issues) can be found here.