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 $1/\zeta(s)$.
$$ $$
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.
Best Answer
Number theorists study a range of different questions that are loosely inspired by questions related to integers and rational numbers.
Here are some basic topics:
Distribution of primes: The archetypal result here is the prime number theorem, stating that the number of primes $\leq x$ is asymptotically $x/\log x$. Another basic result is Dirichlet's theorem on primes in arithmetic progression. More recently, one has the results of Ben Green and Terry Tao on solving linear equations (with $\mathbb Z$-coefficients, say) in primes. Important open problems are Goldbach's conjecture, the twin prime conjecture, and questions about solving non-linear equations in primes (e.g. are there infinitely many primes of the form $n^2 + 1$). The Riemann hypothesis (one of the Clay Institute's Millennium Problems) also fits in here.
Diophantine equations: The basic problem here is to solve polynomial equations (e.g. with $\mathbb Z$-coefficients) in integers or rational numbers. One famous problem here is Fermat's Last Theorem (finally solved by Wiles). The theory of elliptic curves over $\mathbb Q$ fits in here. The Birch-Swinnerton-Dyer conjecture (another one of the Clay Institute's Millennium Problems) is a famous open problem about elliptic curves. Mordell's conjecture, proved by Faltings (for which he got the Fields medal) is a famous result. One can also study Diophantine equations mod $p$ (for a prime $p$). The Weil conjectures were a famous problem related to this latter topic, and both Grothendieck and Deligne received Fields medals in part for their work on proving the Weil conjectures.
Reciprocity laws: The law of quadratic reciprocity is the beginning result here, but there were many generalizations worked out in the 19th century, culminating in the development of class field theory in the first half of the 20th century. The Langlands program is in part about the development of non-abelian reciprocity laws.
Behaviour of arithmetic functions: A typical question here would be to investigate behaviour of functions such as $d(n)$ (the function which counts the number of divisors of a natural number $n$). These functions often behave quite irregularly, but one can study their asymptotic behaviour, or the behaviour on average.
Diophantine approximation and transcendence theory: The goal of this area is to establish results about whether certain numbers are irrational or transcendental, and also to investigate how well various irrational numbers can be approximated by rational numbers. (This latter problem is the problem of Diophantine approximation). Some results are Liouville's construction of the first known transcendental number, transcendence results about $e$ and $\pi$, and Roth's theorem on Diophantine approximation (for which he got the Fields medal).
The theory of modular (or more generally automorphic) forms: This is an area which grew out of the development of the theory of elliptic functions by Jacobi, but which has always had a strong number-theoretic flavour. The modern theory is highly influenced by ideas of Langlands.
The theory of lattices and quadratic forms: The problem of studying quadratic forms goes back at least to the four-squares theorem of Lagrange, and binary quadratic forms were one of the central topics of Gauss's Disquitiones. In its modern form, it ranges from questions such as representing integers by quadratic forms, to studying lattices with good packing properties.
Algebraic number theory: This is concerned with studying properties and invariants of algebraic number fields (i.e. finite extensions of $\mathbb Q$) and their rings of integers.
There are more topics than just these; these are the ones that came to mind. Also, these topics are all interrelated in various ways. For example, the prime counting function is an example of one of the arithmetic functions mentioned in (4), and so (1) and (4) are related. As another example, $\zeta$-functions and $L$-functions are basic tools in the study of primes, and also in the study of Diophantine equations, reciprocity laws, and automorphic forms; this gives a common link between (1), (2), (3), and (6). As a third, a basic tool for studying quadratic forms is the associated theta-function; this relates (6) and (7). And reciprocity laws, Diophantine equations, and automorphic forms are all related, not just by their common use of $L$-functions, but by a deep web of conjectures (e.g. the BSD conjecture, and Langlands's conjectures). As yet another example, Diophantine approximation can be an important tool in studying and solving Diophantine equations; thus (2) and (5) are related. Finally, algebraic number theory was essentially invented by Kummer, building on old work of Gauss and Eisenstein, to study reciprocity laws, and also Fermat's Last Theorem. Thus there have always been, and continue to be, very strong relations between topics (2), (3), and (8).
A general rule in number theory, as in all of mathematics, is that it is very difficult to separate important results, techniques, and ideas neatly into distinct areas. For example, $\zeta$- and $L$-functions are analytic functions, but they are basic tools not only in traditional areas of analytic number theory such as (1), but also in areas thought of as being more algebraic, such as (2), (3), and (8). Although some of the areas mentioned above are more closely related to one another than others, they are all linked in various ways (as I have tried to indicate).
[Note: There are Wikipedia entries on many of the topics mentioned above, as well as quite a number of questions and answers on this site. I might add links at some point, but they are not too hard to find in any event.]