定义:
定义两个数论函数(f)、(g) 的 Dirichlet 卷积为:
[left ( f*g ight )left ( n ight )=sum_{d|n}fleft ( d ight )gleft ( frac{n}{d} ight )]
性质:
Dirichlet卷积满足交换律,结合律,分配律
[left ( f*g ight )*h=f*left ( g*h ight )]
[f*g=g*h]
[left ( f+g ight )*h=f*h+g*h]
其中(varepsilon ) 是Dirichlet卷积的单位元(任何函数卷(varepsilon ) 都是其本身)
[varepsilon = mu *1Leftrightarrow varepsilon left ( n ight )=sum_{d|n}mu left ( d ight )]
[d=1*1Leftrightarrow dleft ( n ight )=sum_{d|n}1]
[sigma =d*1Leftrightarrow sigma left ( n ight )=sum_{d|n}d]
[varphi =mu *IDLeftrightarrow varphi left ( n ight )=sum_{d|n}dcdot mu left ( frac{n}{d} ight )]