GTM 84 - Ch 2 - Application of Unique Factorization

It seems that we always use UF (distribution of factors) to consider the relation "|" , "<"(approximation of \(\pi(x), \theta(x)\)), represent the natural numbers(\(\sum1/p\) diverges, Mobius Inversion, Dirichlet Product).

1 Results

Notes: we consider UFD, "1" stands for units group.

1.1 In the Ring \(\mathbb{Z}\) there are infinitely many prime numbers.

• Euclide's method

• Double counting method

Notes: \(\mathbb{Z}_p\)(all the rational numbers \(a/b, \ p\not|\ b\)) is UFD, but there is only one prime, all primes are associate

\(\mathbb{Z}_p\): \(a/b, \ p \not | \ b,\ a,\ b\in \mathbb{Z}\)

1.2 Arithmetic Functions

A Mobius Inversion Theorem(\(I \circ\mu=\mathbb{I}\))

Let \(F(n)=\sum_{d|n}f(d)\), Then \(f(n)=\sum_{d|n}F(n/d)\mu(d)\)

It is easy understand to use Dirichlet Product!

Notes:

• We can give a more general form if we think it from combinatorics

• We can give another proof for "\(\phi(n)=n\prod(1-\frac{1}{p_i})\)" using Inversion

B Dirichlet Product

Let A={mutiplicative function}

(A, Dirichlet Product) form a group.

1.3 \(\sum1/p\) Diverges

A \(\sum\frac{1}{p}\) diverges

B \(\sum q^{-deg\ p(x)}\) diverges

proof:

\(p_i(x)\) denote the monic irreducible polynomial

\[\lambda(n)=\prod_{deg\ p_i(x)<n} (1-q^{-deg\ p_i(x)})^{-1} \]

It is easy to find \(\lambda(n)\) diverges, and

\[\log\lambda(n)<\sum_{deg\ p_i(x)<n} q^{-deg\ p_i(x)} +\sum_{deg\ p_i(x)<n} q^{-2deg\ p_i(x)} \]

So it makes sense.

1.4 The Growth of \(\pi(x)\)

\[\log\log(x)\le\pi(x) \]

• Eulcide

\[\log 2\cdot\log(x/2)\le\pi(x) \]

• square-free counting

\[\theta(x)=\sum_{p\le x}\log p\\ \theta(x)<(4\log2)x \]

• Consider the factor

\[c_1\frac{x}{\log x}\le\pi(x)\le c_2\frac{x}{\log x} \]

• Utilize the bounds of \(\theta(x)\) and the relation between \(\pi(x)\) and \(\theta(x)\); analysize the factors of \(\left( \begin{array}{c} 2n \\ n \end{array} \right)\)