I wrote this to full my "expository paper" requirement as a sophomore at Harvard (Fall, 1999). I feel that this paper represents in some sense the culmination of my "early years" as a mathematician; in other words, it was written before I really knew anything "deep and structural". Nonetheless, there are many new results and methods, and this paper would be a fun read for anyone who--like me when I was in college-- loves bizarre and wild series evaluations and formulae. The paper is very much in the spirit of Ramanujan.

The official abstract says: Using the theory of residues and Fourier Analysis, we derive many examples of modular forms, including the Dedekind Eta function. After proving several results of Ramanujan, we introduce a function that generalizes $\sech(\sqrt{1/2\pi}z)$ in that it is its own Fourier Transform and reduces to the above as a special case. We show how this leads to a proof of the functional equation for $L(s,\chi)$, and then generalize several results of Ramanujan on self-reciprocal functions. Finally, we show how contour integration may be used to derive explicit product representations for j functions of particular (genus 0) subgroups of GL_2(Z).

In actuality, this paper does quite a bit more than this, and even establishes a kind of analytic continuation formula for continuous analogues of multiple-zeta functions (Euler-Zagier sums) that generalizes a formula of Zagier.

I submitted the paper to The Ramanujan Journal in 2000, and the Referee suggested that I split it up into several papers exploring some of the ideas therein. I unfortunately never got around to doing this as college became rather occupying. One day I'll come back to some of these ideas from a more sophisticated point of view.