Links between several complex variables and number theory?

Question

Can several complex variables be used to prove results in number theory?

If there exists such a proven result, please refer me to the paper/article..

Any help would be appreciated.

Answer 1

There are many applications within the theory of automorphic forms. I think perhaps the most understandable comes from the theory of Multiple Dirichlet series --- i.e. series of the form $$ \sum_{m, n} \frac{a(m, n)}{m^s n^w}.$$

Sometimes these objects arise naturally. For instance, the Dirichlet series associated to a weight $1/2$ Eisenstein series $E(z, w)$ on $\textrm{GL}(2)$ looks like $$ \sum_{n} \frac{L(w, \chi_n) P(n, w)}{n^s},$$ where $\chi_n$ is a Dirichlet character defined mod $n$ and where $P(n, w)$ is a finite correction polynomial when $n$ is not squarefree (and is $1$ otherwise). This is a Dirichlet series whose coefficients are themselves Dirichlet series. This was a major area of investigation by Hoffstein, Cinta, Bump, Friedberg, and Brubaker starting around 1990.

Hoffstein and Hulse recently showed how to use multiple Dirichlet series (and the spectral theory of automorphic forms) to understand a broad class of shifted convolution sums of the form $$ \sum_{n, m \geq 1} \frac{a(n)b(n+m)}{(n+m)^s},\tag{1}$$ where $a(n)$ and $b(n)$ are coefficients of automorphic forms. The interesting thing is that they wanted to understand the meromorphic continuation of $(1)$, but the only way there were able to do that was to introduce an auxiliary variable $w$ and understand the meromorphic continuation to $\mathbb{C}^2$ of $$ \sum_{n, m \geq 1} \frac{a(n) b(n+m)}{(n+m)^s m^w}.$$

If you look up these papers and the papers that cite them, you'll find a large literature of people applying the theory of multiple complex variables to (analytic) number theory.

Answer 2

If two counts as "several", the most famous example is to take squared moduli of $(a+bi)(c+di)=ac-bd+(ad+bc)i$ to prove sums of two squares are closed under multiplication.