[Scribbled notes] on number theory function

Number theory GPBH, so open pit

Common arithmetic function

Mobius function $ \ mu $

1. Definitions:

• $ \ Mu (1) = 1 $
• If there is no $ D $ square factor, $ \ mu (d) = (-1) ^ k $, $ k $ $ D $ is the number of prime factors
• Otherwise, $ \ mu (d) = 0 $

2. Nature:

• For any positive integer $ n $, there $ \ sum \ limits_ {d | n} \ mu (d) = [n = 1] $.
• $ \ Mu $ is a multiplicative function.
• $\sum \limits _{d|n}\frac{\mu(d)}{d}=\frac{\phi(n)}{n}$

Euler function $ \ varphi $

1. Definitions:

• $\varphi (n)= \sum \limits _{i=1}^{n} [gcd(i,n)=1]$

2. Nature:

• $\sum \limits _{d|n} \varphi (d)=n$
• $\varphi(n) = n * \prod (1 - \frac{1}{p_i})$
• $a^{\varphi(m)} \equiv 1 \pmod {m}$
• For $ n = p ^ k $, there $ \ varphi (n) = (p - 1) * p ^ {k - 1} $
• Multiplicative function

Number divisors D $ () $

1. Definitions:

• RT

2. Properties: multiplicative function

About the number and $ \ sigma () $

1. Definitions:

• RT

2. Properties: multiplicative function

Metafunction $ \ epsilon $

1. Definitions:

• $\epsilon(n)=[n=1]$

2. Nature:

• For any multiplicative function $ f $, there is $ f * \ epsilon = f $.
• Multiplicative function

Identity function $ I () $

1. Definitions:

• $ I (n) $ always 1.

2. Nature:

• Multiplicative function

Unit functions $ id () $

1. Definitions:

• $id(n)=n$

2. Nature:

• Multiplicative function

 

To Be Continued.