Mobius Inversion learning notes, Mobius Inversion

Mobius Inversion learning notes, Mobius Inversion

It's really hard for mathematics. Orz JZP, Orz popopoqqq, Orz ......

First, we will introduce a front-end thing ..

Product Functions

For functions defined on N +, if there is at that time, it is a product function. In particular, if it is true at any time, it is a fully product function.

Obviously, the product of the product function is also the product function.

Mobius function:

Defined

Obviously, the Mobius function is a product function.

Nature:

Proof omitted .. Go to number theory ..

Mobius Inversion: if it is satisfied, there will be; If yes, the opposite will also be true ..

Proof:

The coefficient of the allocation rate is obtained from step 2 to step 3, and step 3 to Step 4 only when = 1 is = n = 1.

The Mobius Inversion can be used to simplify some operations, such as the logarithm of a and B, which requires a and B, which is difficult to calculate. If there is a logarithm of a and B, it is easy to obtain it according to the multiplication principle, and so it can be obtained by using Mobius Inversion, so that only a multiple of the enumerated n can be calculated ..

In general, it can be considered to include only the count of the property n, but all the counts containing the property n, so that is, the sum, which is generally easier to obtain, therefore, we can use the Mobius Inversion to convert the problem to the problem, so as to simplify the operation.

This is just my simple understanding.