[Knowledge] some of number theory

• Arithmetical functions, multiplicative function

Codomain: arbitrarily set value range comprising

Arithmetical functions: domain is a positive integer, accompany domain is a complex function

　　Multiplicative function: For arbitrary coprime integers a, b with a f (ab) = f (a ) f (b).

　　Completely multiplicative function: For any integers a, b with a f (ab) = f (a) f (b).　　

　　　　Common multiplicative function:

　　　　　　$[Phi]\; (Euler\; function,\; calculates\; the\; number\; n\; positive\; integers\; coprime)$

　　　　　　[mu] (Mobius function, the number of non-square of the number of prime factors)

$[sigma]\; (n\; and\; all\; of\; the\; n-factor)$

$D\; (n-n\; of\; the\; number\; of\; factors)$

　　　　Common totally multiplicative function:

　　　　　　[epsilon] (defined as: If n = 1, ε (n) = 1; if n> 1, ε (n) = 0 respectively referred to as "for Dirichlet convolution multiplication unit")

　　　　　　I (the same function, is defined as 1 (n) = 1)

$id\; (function\; units,\; defined\; as\; Id\; (n)\; =\; n)$

 

• $Dirichlet\; convolution$

$Set\; f,\; g\; are\; two\; arithmetic\; functions,\; set\; F\; =\; H*G,\; which\; is\; the\; Dirichlet\; convolution:$

 

　　Properties: commutative, associative law, and distribution ratio, (if both multiplicative function, but also as a multiplicative function convolution)

　　Identity element: [epsilon]

　　Some properties of Dirichlet convolution binding obtained: