# Appendix D. Real and Complex Analysis Review

## D.1 Real Numbers

The real numbers $\mathbb{R}$ extend the rational numbers $\mathbb{Q}$ by filling gaps such as

$$
\sqrt{2},\quad \pi,\quad e.
$$

They form an ordered field. This means addition, multiplication, and order are compatible. For example, if

$$
a<b,
$$

then

$$
a+c<b+c.
$$

If

$$
0<c,
$$

then

$$
ac<bc.
$$

The real numbers are also complete: every nonempty set of real numbers that is bounded above has a least upper bound. This property separates $\mathbb{R}$ from $\mathbb{Q}$, and it is the foundation of limits, continuity, integration, and analytic number theory.

## D.2 Sequences

A sequence of real numbers is a function

$$
a:\mathbb{N}\to\mathbb{R}.
$$

We usually write its values as

$$
a_1,a_2,a_3,\ldots
$$

A sequence $(a_n)$ converges to $L$ if, for every $\varepsilon>0$, there exists $N$ such that

$$
n\ge N \implies |a_n-L|<\varepsilon.
$$

We then write

$$
\lim_{n\to\infty}a_n=L.
$$

In number theory, sequences appear as arithmetic functions, partial sums, convergents of continued fractions, coefficients of modular forms, and values of $L$-functions.

## D.3 Series

An infinite series is an expression

$$
\sum_{n=1}^{\infty}a_n.
$$

It converges if the sequence of partial sums

$$
s_N=\sum_{n=1}^{N}a_n
$$

has a finite limit.

The harmonic series

$$
\sum_{n=1}^{\infty}\frac{1}{n}
$$

diverges, while the $p$-series

$$
\sum_{n=1}^{\infty}\frac{1}{n^p}
$$

converges exactly when

$$
p>1.
$$

This fact is central to the first analytic study of primes through the zeta function.

## D.4 Absolute Convergence

A series

$$
\sum_{n=1}^{\infty}a_n
$$

converges absolutely if

$$
\sum_{n=1}^{\infty}|a_n|
$$

converges.

Absolute convergence is stronger than ordinary convergence. It permits rearrangement of terms and supports products of series.

Euler products rely on absolute convergence in some half-plane. For instance, the product formula

$$
\zeta(s)=\prod_{p}\frac{1}{1-p^{-s}}
$$

is first justified when the real part of $s$ is greater than $1$.

## D.5 Functions and Limits

For a function

$$
f:\mathbb{R}\to\mathbb{R},
$$

we say

$$
\lim_{x\to a}f(x)=L
$$

if $f(x)$ becomes arbitrarily close to $L$ whenever $x$ is sufficiently close to $a$.

Limits describe local behavior. They are used to define derivatives, continuity, asymptotic notation, and analytic continuation.

A function is continuous at $a$ if

$$
\lim_{x\to a}f(x)=f(a).
$$

Continuity allows local approximation and global control.

## D.6 Differentiation

The derivative of $f$ at $a$ is

$$
f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h},
$$

when this limit exists.

Derivatives measure local rate of change. In analytic number theory, differentiation appears in estimates, generating functions, contour integrals, logarithmic derivatives, and explicit formulae.

The logarithmic derivative

$$
\frac{f'(s)}{f(s)}
$$

is especially important. For the zeta function, it connects zeros and primes.

## D.7 Integration

The integral

$$
\int_a^b f(x)\,dx
$$

measures accumulated value. In number theory, integration is often used to approximate sums.

A basic example is the comparison between

$$
\sum_{n\le x} f(n)
$$

and

$$
\int_1^x f(t)\,dt.
$$

This idea underlies partial summation, Tauberian arguments, estimates for arithmetic functions, and analytic approximations to counting functions.

## D.8 Asymptotic Notation

Asymptotic notation describes growth.

We write

$$
f(x)=O(g(x))
$$

if there exist constants $C>0$ and $x_0$ such that

$$
|f(x)|\le C|g(x)|
$$

for all $x\ge x_0$.

We write

$$
f(x)\sim g(x)
$$

if

$$
\lim_{x\to\infty}\frac{f(x)}{g(x)}=1.
$$

For example, the prime number theorem states

$$
\pi(x)\sim \frac{x}{\log x}.
$$

This means the ratio of the two quantities tends to $1$, not that their difference is small.

## D.9 Complex Numbers

The complex numbers are

$$
\mathbb{C}=\{a+bi:a,b\in\mathbb{R},\ i^2=-1\}.
$$

The real part of $z=a+bi$ is

$$
\operatorname{Re}(z)=a.
$$

The imaginary part is

$$
\operatorname{Im}(z)=b.
$$

The complex conjugate is

$$
\overline{z}=a-bi.
$$

The absolute value, or modulus, is

$$
|z|=\sqrt{a^2+b^2}.
$$

Complex numbers are essential in analytic number theory because zeta functions and $L$-functions are naturally complex functions.

## D.10 Complex Exponential

Euler’s formula says

$$
e^{i\theta}=\cos\theta+i\sin\theta.
$$

Thus every nonzero complex number can be written in polar form:

$$
z=re^{i\theta},
$$

where

$$
r=|z|.
$$

This representation connects multiplication with rotation and scaling.

Complex exponentials appear in Fourier analysis, additive characters, Gauss sums, modular forms, and the circle method.

## D.11 Holomorphic Functions

A function

$$
f:U\to\mathbb{C}
$$

on an open set $U\subseteq\mathbb{C}$ is holomorphic if it is complex differentiable at every point of $U$.

Holomorphic functions are far more rigid than real differentiable functions. They have power series expansions, obey strong maximum principles, and are controlled by their values on small sets.

This rigidity is one reason complex analysis is so effective in number theory.

## D.12 Power Series

A power series has the form

$$
\sum_{n=0}^{\infty}a_n(z-z_0)^n.
$$

It converges inside a disk

$$
|z-z_0|<R
$$

and defines a holomorphic function there.

Power series appear in generating functions, modular forms, $q$-expansions, local zeta functions, and formal group laws.

For example, modular forms are often studied through expansions of the form

$$
f(q)=\sum_{n=0}^{\infty}a_nq^n.
$$

The coefficients $a_n$ frequently contain arithmetic information.

## D.13 Meromorphic Functions

A meromorphic function is holomorphic except at isolated poles.

A pole at $z_0$ means that near $z_0$, the function behaves like

$$
\frac{a_{-m}}{(z-z_0)^m}
$$

plus less singular terms.

The Riemann zeta function is meromorphic on $\mathbb{C}$. It has a simple pole at

$$
s=1.
$$

This pole is one of the analytic sources of the density of prime numbers.

## D.14 Analytic Continuation

Analytic continuation extends a holomorphic function beyond its original domain when the extension is forced by local agreement.

The zeta function is first defined by

$$
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}
$$

for

$$
\operatorname{Re}(s)>1.
$$

Complex analysis extends it to a meromorphic function on the whole complex plane.

This extension allows the study of values and zeros outside the region where the original series converges.

## D.15 Residues

If a meromorphic function has a Laurent expansion

$$
f(z)=\sum_{n=-m}^{\infty}a_n(z-z_0)^n
$$

near $z_0$, then the residue of $f$ at $z_0$ is

$$
\operatorname{Res}_{z=z_0} f(z)=a_{-1}.
$$

Residues measure the coefficient of the simple pole term.

The residue theorem turns contour integrals into sums of local data at singularities. In analytic number theory, this connects sums over integers, sums over primes, and sums over zeros of analytic functions.

## D.16 Contour Integration

A contour integral integrates a complex function along a path in the complex plane.

If $f$ is holomorphic inside and on a closed contour $\gamma$, then

$$
\int_{\gamma} f(z)\,dz=0.
$$

If $f$ has isolated singularities inside $\gamma$, then the residue theorem gives

$$
\int_{\gamma} f(z)\,dz =
2\pi i \sum \operatorname{Res}(f).
$$

Contour integration is a principal tool in proving explicit formulae and asymptotic estimates.

## D.17 Fourier Analysis

Fourier analysis decomposes functions into oscillatory components.

On the circle, one studies expansions such as

$$
f(x)\sim \sum_{n\in\mathbb{Z}} c_n e^{2\pi i n x}.
$$

In number theory, Fourier analysis appears through additive characters:

$$
e^{2\pi i n x}.
$$

It is central to the circle method, equidistribution, modular forms, Poisson summation, and automorphic forms.

## D.18 Poisson Summation

The Poisson summation formula relates a sum over a lattice to a sum over the dual lattice. In one common form,

$$
\sum_{n\in\mathbb{Z}} f(n) =
\sum_{m\in\mathbb{Z}} \widehat{f}(m),
$$

where $\widehat{f}$ is the Fourier transform of $f$.

This formula is a bridge between discrete arithmetic and continuous analysis. It appears in theta functions, modularity, lattice point counting, and analytic estimates.

## D.19 Analytic Tools in Number Theory

| Analytic Concept | Number-Theoretic Use |
|---|---|
| Sequences | arithmetic functions, recurrence, approximations |
| Series | zeta functions, Dirichlet series |
| Infinite products | Euler products, prime factorization |
| Differentiation | logarithmic derivatives, estimates |
| Integration | summation formulae, counting functions |
| Complex analysis | analytic continuation, residues, zeros |
| Fourier analysis | characters, modular forms, circle method |
| Asymptotics | growth rates, error terms |

Analysis gives number theory a language for size, density, approximation, and oscillation. Classical arithmetic asks which integers exist. Analytic number theory asks how often they occur and how regularly they are distributed.