holomorphicity

A function $f:\mathbb C\rightarrow\mathbb C$ is holomorphic at the point $z\in\mathbb C$ if the limit

$$\lim_{h\to 0}\frac{f(z+h)-f(z)}{h}(h\in\mathbb C)$$

exists.

characteristic properties of holomorphic functions

• Contour integration: If f is holomorphic in $\Omega$,then for appropriate closed paths in $\Omega$

  $$\int_\gamma f(z)dz=0$$

• Regularity: If f is holomorphic, then f is indefinitely differentiable.

• Analytic continuation: If f and g are holomorphic functions in $\Omega$ which are equal in an arbitrarily small disc in $\Omega$,then f=g everywhere in $\Omega$.

• The zeta function

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

• The theta function

  $$\Theta(z|\tau)=\sum_{n=-\infty}^{\infty}e^{\pi in^2}e^{2\pi inz}$$

• Basic properties

The complex plane

Commutativity

Associativity

Distributivity

The absolute value

The triangle inequality

The complex conjugate

In polar form

The prime number theory

$$\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1$$ $$\pi(x)\ is\ the\ number\ of\ primes\ between\ 1\ and\ x$$

Lemma 1

Every nonzero integer can be written as a product of primes.

Theorem 1

For every nonzero integer n there is a prime factorization

$$n=(-1)^{\epsilon(n)} \prod_{p}p^{a(p)}$$

with the exponents uniquely determined by n.

$$a(p)=ord_{p}n$$

Lemma 2

If$a,b\in \mathbb{Z}$ and b>0, there exist q,r\in\mathbb{Z}such that a=qb+r with $0\leq r<b$

Definition

If $a_1, a_2,...,a_n\in\mathbb{Z}$, we define $(a_1, a_2,..., a_n)$ to be the set of all integers of the form $a_1x_1+a_2x_2++a_nx_n$ with $x_1, x_2, \cdots, x_n\in \mathbb Z$.

Lemma 3

If $a,b\in\mathbb Z$, then there is a $d\in\mathbb Z$ such that (a,b)=(d).

Definition

Let $a,b\in\mathbb Z$. An integer d is called a greatest common divisor of a and b if d is a divisor of both a and b and if every other common divisor of a and b divides d.

Lemma 4

Let $a,b\in\mathbb Z$.If (a,b)=(d) then d is a greatest common divisor of a and b.

Definition

We say that two integers a and b are relatively prime if the only common divisors are $\pm 1$, the units.

Proposition 1.1.1

Suppose that a|bc and that (a,b)=1.Then a|c.

Corollary 1

If p is a prime and p|bc, then either p|b or p|c.

Corollary 2

Suppose that p is a prime and that $a,b\in\mathbb Z$. Then $ord_pab=ord_pa+ord_pb$.

Unique Factorization in k[x]

Reference book
A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen