Möbius inversion Study Notes

Maths is so hard, Orz Jzp,orz Popoqqq,orz ...

Let's introduce a front-facing thing.

integrable function

For a function defined on a n+, if there is one, it is an integrable function. In particular, if it is established at all times, it is a fully integrable function.

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

Möbius function:

Defined as

Obviously, the Möbius function is an integrable function.

Properties:

Proof slightly. Turn to the book of number theory.

Möbius inversion: The setting satisfies, then has, when, has, the contrary also establishes.

Prove:

The second to the third pass the coefficient of the allocation factor comes, the third step to the fourth step is because only when =1 is =n = 1.

Möbius inversion can be used to simplify some operations, such as the representation of a B logarithm, which is a A, a, b two, not good calculation. If there is the representation of a, B logarithm, according to the multiplication principle is easy to obtain, but also has, so can be used Möbius inversion, so long as the enumeration n multiples of a quantity can be calculated.

Generally speaking, it can be considered to contain only the nature of the count of N, but all contains the nature of the count of N, so that is, and, it is generally easier to obtain, then can be used to Möbius inversion of the problem to be solved, in order to simplify the operation.

It's just my simple comprehension, and (Word's formula comes up with a really painful egg

Möbius inversion Study Notes