A summary of "the Mo-in-the-black inversion" project

First, define the definition field of the number theory function as a function of f (1) Not equal to 0 under the positive integer field.

From Syu Gau

http://www.zhihu.com/question/23764267/answer/26007647

There are several concepts

1, convolution:
is set to two number theory functions (that is, a complex value function that defines a field with a set of natural numbers), the convolution operation is defined as

It can be proved that convolution operation satisfies:
1) Exchange Law:
By definition clearly.

2) Binding law:
Examine both sides of the action on the left is

to the right.

So the two sides are equal.

3) Existence of unit element makes
We need

It is not difficult to guess what should be defined as
In fact, direct validation can be


The above shows that the number theory function forms a commutative group under convolution meaning.


2, multiplication unit element
The above is the unit element in the convolution sense of the number theory function, and the unit element in the sense of ordinary multiplication is obviously the function that all natural numbers are mapped to 1.


3, the Mohs function
The inverse element in convolutional sense is called the Möbius function. Which means satisfying

The only number-theoretic function.
Write this expression open
............ (*)

Typically, the Möbius function is defined as
；
, if it can be written as a product of different primes;
, in other cases.

According to this definition it is not difficult to prove (*) formula.
For, (*) Form;
For, with the basic theorem of arithmetic to write

So




Now, what is Möbius's anti-speech?

When and only if

In other words,


Prove:

Instead

In the case of GCD, we assume that G (i) represents F (i) under I=GCD (x, y) in i|gcd (x, y) with F (n) =σg (d) d|ng (n) =σf (d) *u (N/D) d|n which is essentially a repulsion ~

A summary of "the Mo-in-the-black inversion" project