Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it.

$\textbf{Background}$

I am an undergraduate student and I'm very interested in number theory. I've tried to study alone some algebraic number theory and I believe that I know a lot of things about algebraic number fields and $p$-adic numbers, all using Neukirch's ANT book.

$\textbf{Question}$

I try to read a lot about number theory on the internet and I realize that a lot of complex analysis stuff is used. My question is basically: how does these complex analysis stuff, like Riemann Zeta function, relate explicitly to number theory? I asked a long time ago about the use of complex numbers in number theory and people told me about some results that used Gaussian Integers, for example to study primes that can be written as a sum of two squares in $\mathbb{Z}$, but I believe this fits more in the study of algebraic number fields and it is far away from the Riemann Zeta stuff.

Can someone give me sort of intuition or motivation on how does these complex analysis ideas relate to a problems involving integers or number theory in general? Thank you in advance.

Solution 1:

Note: The following touches only a few aspects far from being representative for the wide connection of complex analysis with number theory.

General Note: When asking for connections of complex analysis with number theory you should delve into analytic number theory. This branch of number theory is roughly divided into additive number theory and multiplicative number theory.

From T. Apostols introductory section of his classic Modular Functions and Dirichlet Series in Number Theory:

Additive number theory is concerned with expressing an integer $n$ as sum of integers from some given set $S$. ...

Let $f(n)$ denote the number of ways $n$ can be written as a sum of elements of $S$. We ask for various properties of $f(n)$, such as its asymptotic behavior for large $n$. In a later chapter we will determine the asymptotic value of the partition function $p(n)$ which counts the number of ways $n$ can be written as a sum of positive integers $\leq n$.

The partition function $p(n)$ and other functions of additive number theory are intimitely related to a class of functions in complex analysis called elliptic modular functions. They play a role in additive number theory analogous to that played by Dirichlet series in multiplicative number theory. ...

We see from his intro that complex analysis plays a key role and also that the asymptotic study of function is essential for gaining insight.

In order to get information about the behavior of numerical sequences $a_n$ for large $n$, we study the corresponding generating functions $F(z)$ \begin{align*} F(z)=\sum_{n=0}^{\infty}a_nz^n \end{align*} as function of a complex variable $z$. Crucial for the asymptotic behavior is the behavior of the function near its singularities.

Asymptotic Behavior: P. Flajolet and R. Sedgewick explain in Analytic Combinatorics this as follows

Comparatively little benefit results from assigning only real values to the variable $z$ that figures in a univariate generating function. In contrast, assigning complex values turns out to have serendipitous consequences. ...

When we do so, a generating function becomes a geometric transformation of the complex plane. This transformation is very regular near the origin—one says that it is analytic (or holomorphic). In other words, near $0$, it only effects a smooth distortion of the complex plane. Farther away from the origin, some cracks start appearing in the picture. These cracks—the dignified name is singularities—correspond to the disappearance of smoothness. It turns out that a function’s singularities provide a wealth of information regarding the function’s coefficients, and especially their asymptotic rate of growth. Adopting geometric point of view for generating functions has a large pay-off.

By focusing on singularities, analytic combinatorics treads in the steps of many respectable older areas of mathematics. For instance, Euler recognized that for the Riemann zeta function $\zeta(s)$ to become infinite (hence have a singularity) at $1$ implies the existence of infinitely many prime numbers; Riemann, Hadamard, and de la Vall'ee-Poussin later uncovered deep connections between quantitative properties of prime numbers and singularities of $1/\zeta(s)$.

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Application of complex analysis: Going back to Apostol's book we can find a famous application of the elliptic modular functions, namely Rademacher's series for the partition function:

From chapter 5:

... The unrestricted partition function $p(n)$ counts the number of ways a positive integer $n$ can be expressed as a sum of positive integers $\leq n$. The number of sums is unrestricted, repetition is allowed, and the order of the summands is not taken into account.

The partition founction is generated by Euler's infinite product \begin{align*} F(x)=\prod_{m=1}^{\infty}\frac{1}{1-x^m}=\sum_{n=0}^{\infty}p(n)x^n, \end{align*} where $p(0)=1$. Both the product and series converge absolutely and represent the analytic function $F$ in the unit disk $|x|<1$.

...

The partition function $p(n)$ satisfies the asymptotic relation \begin{align*} p(n)\sim\frac{e^{K\sqrt{n}}}{4n\sqrt{3}}\qquad \text{as } n\rightarrow \infty, \end{align*} where $K=\pi(2/3)^{1/2}$. This was first discovered by Hardy and Ramanujan in 1918 ...

By integrating along a path in the complex plane and considering thereby so-called Ford circles for Farey series (you may want to look at these interesting objects) Rademacher found the following celebrated representation of $p(n)$ for $n\geq 1$ as convergent series:

Apostol (Theorem 5.10): If $n\geq 1$ the partion function $p(n)$ is represented by the convergent series

\begin{align*} p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}A_k(n)\sqrt{k}\frac{d}{dn} \left(\frac{\sinh\left\{\frac{\pi}{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right\}}{\sqrt{n-\frac{1}{24}}}\right) \end{align*} where \begin{align*} A_k(n)=\sum_{{0\leq h<k}\atop{(h,k)=1}}e^{\pi i s(h,k)-2\pi i nh/k} \end{align*}

and with $s(h,k)$ being the Dedekind sum \begin{align*} s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\left(\frac{hr}{k}-\left[\frac{hr}{k}\right]-\frac{1}{2}\right) \end{align*}

Here is a last indication how important the study of the singularities in the complex plane of a function is.

Riemann's Zeta Function:

H.M.Edwards discusses in the first chapter of his classic Riemann's Zeta Function Riemann's epoch-making 8-page paper On the number of primes less than a given Magnitude.

From section 1.10 The Product representation of $\zeta(s)$:

A recurrent theme in Riemann's work is the global characterization of analytic function by their singularities. Since the function $\log\zeta(s)$ has logarithmic singularities at the roots $\rho$ of $\zeta(s)$ and no other singularities, it has the same singularities as the formal sum \begin{align*} \sum_{p}\log\left(1-\frac{s}{p}\right)\tag{1}. \end{align*} Thus if this sum converges and if the function it defines is in some sense as well behaved near $\infty$ as $\log \zeta(s)$ is, then it should follow that the sum (1) differs from $\log \zeta(s)$ by at most an additive constant; setting $s=0$ gives the value $\log \zeta(0)$ for this constant, and hence exponentiation gives \begin{align*} \zeta(s)=\zeta(0)\prod_s\left(1-\frac{s}{p}\right)\tag{2} \end{align*} as desired. This is essentially the proof of the product formula (2) which Riemann sketches.

Solution 2:

Here are two answers.

(1) Prove that $\pi$ is transcendental. Very brief summary of proof: suppose that $\pi$ is algebraic; then $i\pi$ is also algebraic. Construct a polynomial $h(z)$, with integer coefficients, having roots $\beta_k$ which are related in a certain way to the algebraic conjugates of $i\pi$, and consider the sum of integrals $$J=\sum_k\int_0^{\beta_k}\! z^nh(z)^{n+1}e^{\beta_k-z}\,dz\ .$$ Show that a certain multiple of $J$ is a positive integer less than $n!$, and also that it is a multiple of $n!$. This is a contradiction.

Admittedly, this is a fairly weak example, because we are only integrating entire functions of a complex variable. This means that we can integrate in much the same way as we do for real functions, and we are not using any of the "specifically complex" techniques like CIF or residues. So here is another example.

(2) Investigate the function $r(n)$, the number of ways of writing the positive integer $n$ as an ordered sum of four squares. Note that $r(n)$ is the coefficient of $z^n$ in $$f(z)^4=\Bigl(\sum_{k=0}^\infty z^{k^2}\Bigr)^4\ ,$$ and is therefore the residue at $z=0$ of $$\frac{f(z)^4}{z^{n+1}}\ .$$ Provided this function has no further singularities with $|z|\le a$, we have $$r(n)=\frac1{2\pi i}\int_{|z|=a} \Bigl(\sum_{k=0}^\infty z^{k^2}\Bigr)^4\frac{dz}{z^{n+1}}\ ,$$ and "in principle" the integral can be evaluated in order to find $r(n)$. Of course, there is a lot of work to be done, but I hope that this shows you how certain aspects of complex analysis (specifically, integration by residues) are connected with number theory.