# 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+P_{2}, where p is a prime and P_{2} 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
a^{2}+b^{4}. 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.

### Articles

- S. Baier, T.D. Browning, G. Marasingha,
L. Zhao,
*Averages of shifted convolutions of d*, Proceedings of the Edinburgh Mathematical Society (2)_{3}(n)**55**(2012), no. 3, 551–576. - Gihan Marasingha,
*Almost primes represented by binary forms*, Journal of the London Mathematical Society (2)**82**(2010), 295–316. - Gihan Marasingha,
*On the representation of almost primes by pairs of quadratic forms*, Acta Arithmetica**124**(2006), 327–355. - Gihan Marasingha,
*On pairs of quadratic forms*, DPhil thesis, University of Oxford, 2005. - Gihan Marasingha,
*Elementary Wavelet Analysis*, MSci Dissertation, Imperial College, University of London. - Gihan Marasingha,
*Commutative Banach Algebras and the Gelfand Representation Theorem*, fourth year project, Imperial College, University of London.

- S. Baier, T.D. Browning, G. Marasingha,
L. Zhao,
### Internal Links:

- Partial Fractions Explained — a rigourous analysis of this high school topic.
- Sifting the Primes — a discussion of sieve methods and their application.
- The RSA Code and congruences — an introduction to crytpography via RSA.
- Elementary wavelet analysis — my MSci thesis.

### Encyclopaedias:

- Planetmath — Definitions, theorems, etc.
- Mathworld — a small mathematical encyclopaedia.
- Mathematics archives — an organised collection of topics at the undergraduate level.

### Conferences, etc.

- Numbertheory.org — conferences, lecture notes, etc.

### Journals:

### Online Books and Lecture Notes:

- Online number theory lecture notes — from numbertheory.org.
- Noam D. Elkies — analytic number theory.
- J. S. Milne — algebraic geometry.
- Robert B. Ash — algebra, number theory, analysis.

### Magazines:

- Notices of the American Mathematical Society.
- Seed Magazine — general science.
- Plus Magazine — popular mathematics.

### Teaching Advice:

- How to listen to a maths lecture — from the generally wonderful site of Tom Körner.
- Ten Lessons I Wish I Had Been Taught — by Gian Carlo Rota. PDF available from the AMS.
- Helping Undergraduates Learn to Read Mathematics. — by Ashley Reiter.
- How to Read Mathematics — by Shai Simonson and Fernando Gouvea.
- The Most Common Errors in Undergraduate Mathematics — by Eric Schechter.

### TeX/LaTeX Tips:

- TeX Resources — by A. J. Hildebrand. Various tips, including bibliography conventions.
- Simple Margin Adjustment in LaTeX — by A. G. Marasingha.