[MO] "Learning notes" [UPDATE]

Resources:

[Most have not finished reading, at present mainly read the POPOQQQ that article Orz]

Http://wenku.baidu.com/link?url= Kzzxkk64cfu7sfdejbgknpzpfjzjy1zwnoapggo7tpspv4kjvgakstkpzytg46gjquqnx7nb0merxfs4knd2h5fw7s4ohu1o1-6p16_vbem

Http://wenku.baidu.com/view/77396ebb27d3240c8547ef2e.html?re=view

On the prefix of a class of integrable functions and

Http://vfleaking.blog.uoj.ac/slide/87

"0"

What is inversion:

Suppose there are two functions F and g that satisfy:

F (n ) =∑k a[ n , k]*g (k)

The process known as F to ask G is called inversion.

A

1. Iff(n) is a positive integer field and the range is a complex number, i.e.f:Z+→C , it is said F(n) is the number theory function .
2. Iff(n) Is the number theory function, andF(1)=1 , a positive integer for coprimep,q Yes F(p-q)=f(P)-f(Q) , it is called an integrable function .
3. Iff(n) is an integrable function, and for any positive integerp,q All have F(p-q)=f(P)-f(Q) , it is called a fully integrable function .
5. The prefix of the integrable function and also the integrable function

Two

Form two:

"Möbius function"

1.

• The Möbius function μ(n), which is the inverse of the identity function in the multiplication of the Dirichlet convolution, mu*1=e
• The function of the integrand. Proof: μ (n) =∏μ (Pi^ei), each pi^ei is coprime
• μ(1)=1
  
• μ (n) = ( -1) ^k if n is the product of K different primes (the number of mass factors is 1)
• 0 other conditions

2. Nature:

(1)

Proof: When N!=1, the number of quality factor is 1,i a factor of C (K,i), then:

According to the two-term theorem:

Get proof

(2)

Prove:

i/n to the simplest fraction j/d, then d|n and gcd (j,d) =1

The simplest fraction with D as the denominator has Phi (d)

Total N Fractions

(3)

Proof: Using the above formula, naked on the Mohs inversion F (n) =n,g (n) =phi (n)

3. Ask for Möbius function

Linear sieve

BOOLNotp[n];intP[n],mu[n];voidsieve () {mu[1]=1;  for(intI=2; i<=n-1; i++){        if(!notp[i]) p[++p[0]]=i,mu[i]=-1;  for(intj=1; j<=p[0]&&i*p[j]<=n-1; j + +){            intt=i*P[j]; Notp[t]=1; if(i%p[j]==0) {Mu[t]=0;  Break; } Mu[t]=-Mu[i]; }    }}

Möbius inversion is the principle of tolerance on partial sequence set

[MO] "Learning notes" [UPDATE]