# Mathematics

My mathematical interests lie primarily in analytic number theory. This field leans heavily towards the "problem solving" as opposed to the "structure building" side of things, and its methods may sometimes seem ad hoc. On the other hand, the application of analytic methods has led to some deep and striking results, notably the prime number theorem which states that the number of primes less than x is asymptotic to x / log (x).

The use of analytical tools often produces results which consist of a main term and an error term — the challenge then consists in reducing the size of the error term. In this manner, we can easily(!) prove approximations to great problems such as the Goldbach conjecture and the twin primes conjecture.

Much of my recent work has been in the area of sieve theory, a field which dates back to the 3rd century BC with the sieve of Eratosthenes. A famous result in sieve theory is Chen's Theorem, which states that any even number (greater than 2) can be written as p+P2, where p is a prime and P2 is the product of at most 2 primes. There is also the Friedlander–Iwaniec Theorem, which states that there are infinitely many primes of the form a2+b4. I have been using sieve theory to deduce results about almost-primes represented by quadratic forms. This is of interest because the Schinzel-Sierpinski Hypothesis (which generalises the twin primes conjecture) can be rephrased in terms of the representation of almost primes by polynomials.