"Algorithm topic" integrable function

Reference

★ Talking about the prefix and by Skywalkert of a class of integrable functions

Ningzhi number theory function. pdf

"Integrable function"

The approximate and, prefixes, and convolution functions of the integrand are also integrable functions.

1.F (1) = 1.

2. Nature one: For N=∏pi^ki, there is f (n) =∏f (Pi^ki)

Property Two: For the complete integrable function, there are also F (n) =∏f (PI) ^ki and F (n^k) =f (n) ^k

Common integrable functions:

1.D (n) =σd|n1, which represents the number of factors of n, that is, d=i*i

2.σ (n) =σd|nd, which represents the factor of N and, that is, σ=i*id

3.I (n) = 1, identity function

4.id (n) =n, unit function

5.E (n) =[n=1], meta-function, i.e. f=f*e

6.φ (n) =σ[(n,i) =1]*1, Euler functions

7.μ (n), Möbius function, μ (n) = ( -1) The number of element factors with n ^k,k, with the repetition factor μ=0

"Dirichlet convolution"

Defines the Dirichlet convolution of the two number theoretic functions F,g: (F*g) (n) =σd|nf (d) *g (N/D).

1. Möbius function, E (n) =σd|nμ (d), i.e. E=μ*i.

Möbius inversion, from G=f*i, get f=g*μ.

Proof: f=g*μ=f*i*μ=f*e=f.

i.e. by g (n) =σd|nf (d), f (n) =σd|ng (d) *μ (N/D).

Similarly, by G (n) =σn|df (d), f (n) =σn|dg (d) *μ (d/n).

2. Euler function, n=σd|nφ (d), i.e. Id=φ*i.

by inversion, φ=id*μ, i.e. φ (n)/n=σd|nμ (d)/d.

"The technique of σ-transformation"

The Basic Law (specific mathematics):

1. The distribution law, Σkc*ak=c*σkak, is to propose multipliers unrelated to σ.

2. Binding or separating the conditions of adjacent σ.

3. The Exchange law, that is, the enumeration of σ can be changed in order.

4. General distribution Law, σj,kaj*bk= (ΣAJ) * (ΣBK)

5. Multi-Exchange law, when the adjacent Σ enumeration domain is related, it is necessary to satisfy:

[J∈j] [K∈k (j)]=[k∈k '][j∈j ' (k)]

Usually J=k ' is a set of all integers, and the second is launched according to the control of the double and the nature of P (j,k).

6. Swap the element, that is, the enumerator that replaced Σ.

7. Iverson agreed to turn the end of σ into a condition, such as σi∈ii=σi*[i∈i].

Simplification tips:

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"Algorithm topic" integrable function