2025寒训：寒末上午（数学）Day 1

积性函数

定义

\(\forall a \perp b,\ f(ab)=f(a)\times f(b)\)

常见积性函数

\(id(n)=n\)
\(e(n)=[n=1]\)
\(\varphi(n)=n\prod_{p|n,p\in\mathbb{P}} (1-\frac{1}{p})\)
\(I(n)=1\)
\(\mu(n)=0,n\) 有平方因子$\quad \quad mu(n)=(-1)^{\omega(n)},\ $其它情况

卢卡斯定理

\(C^{n}_{m}\equiv C^{n\ \text{mod}\ p}_{m\ \text{mod}\ p}\times C^{\lfloor n/p \rfloor}_{\lfloor m/p\rfloor}\)

狄利克雷卷积

\((f*g)(n)=\sum\limits_{d|n}f(d)g(\frac{n}{d})\)
可以得出
\(e=\mu *1\)
\(id=\varphi *1\)

莫比乌斯反演

例：
\(\quad \sum\limits^{n}_{i=1}\sum\limits^{m}_{j=1}\gcd(i,j)\)
\(=\sum\limits^{n}_{i=1}\sum\limits^{m}_{j=1}\sum\limits_{d|\gcd(i,j)}d[\gcd(i,j)=d]\)
\(=\sum\limits_{d=1}^{\min(n,m)}d\sum\limits_{i=1,d|i}^{n}\sum\limits_{j=1,d|j}^{m}[\gcd(i,j)=d]\)
令\(i'=i/d,j'=j/d\)
\(=\sum\limits_{d=1}^{\min(n,m)}d\sum\limits_{i'=1}^{\lfloor n/d\rfloor}\sum\limits_{j'=1}^{\lfloor m/d\rfloor}[\gcd(i',j')=1]\)
\(=\sum\limits_{d=1}^{\min(n,m)}d\sum\limits_{i'=1}^{\lfloor n/d\rfloor}\sum\limits_{j'=1}^{\lfloor m/d\rfloor}e(\gcd(i',j'))\)
\(=\sum\limits_{d=1}^{\min(n,m)}d\sum\limits_{i'=1}^{\lfloor n/d\rfloor}\sum\limits_{j'=1}^{\lfloor m/d\rfloor}\sum\limits_{d'|\gcd(i',j')}\mu(d')\)
\(=\sum\limits_{d=1}^{\min(n,m)}d\sum\limits_{d'=1}^{\lfloor n/d\rfloor}\mu(d')\sum\limits_{i'=1}^{\lfloor n/dd'\rfloor}\sum\limits_{j'=1}^{\lfloor m/dd'\rfloor}1\)
\(=\sum\limits_{d=1}^{\min(n,m)}d\sum\limits_{d'=1}^{\lfloor n/d\rfloor}\mu(d')\lfloor n/dd'\rfloor\lfloor m/dd'\rfloor\)
\(=\sum\limits_{T=1}^{\min(n,m)}(\sum\limits_{d|T}d\times\mu(T/d))\lfloor n/dd'\rfloor\lfloor m/dd'\rfloor\)
\(=\sum\limits_{T=1}^{\min(n,m)}\varphi(T)\lfloor n/dd'\rfloor\lfloor m/dd'\rfloor\)