莫比烏斯反演的引入
莫比烏斯反演是數論中的重要內容,在許多情況下能夠簡化運算。我們考慮以下求和函式:
![莫比烏斯反演](/img/8/e73/wZwpmLxITM2kzM3kTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL4UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/5/beb/wZwpmL3cjNycTN0ITMzEzM1UTM1QDN5MjM5ADMwAjMwUzLyEzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/e/cd7/wZwpmL3EjM3gjN0UDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1gzL2YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
我們需要找到和之間的關係。從和函式定義當中,我們可以知道:
![莫比烏斯反演](/img/a/f5c/wZwpmL1UTNyczNzIDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL2AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/8/59a/wZwpmLzUDO2EzM4MjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLzYzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/b/d9a/wZwpmLwATN4kTO3IzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyMzLxUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/e/c40/wZwpmLxUTNzATM0ADN3UzM1UTM1QDN5MjM5ADMwAjMwUzLwQzL3IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/b/f71/wZwpmL3EzMykjM4MjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLzYzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/9/e00/wZwpmLxYDMzMTM3YzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL2czLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/e/917/wZwpmL4UTM2gjMyMTO2UzM1UTM1QDN5MjM5ADMwAjMwUzLzkzL0gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/4/bf2/wZwpmL4cDOxQjM0MDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzAzLyEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
那么:
![莫比烏斯反演](/img/3/218/wZwpmL3IDO0QDMxYzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLygzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/a/a53/wZwpmLxUzMygTN3gDN3UzM1UTM1QDN5MjM5ADMwAjMwUzL4QzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/8/b48/wZwpmLzQDN3gTNzAzN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwczLwMzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/5/4b7/wZwpmLwQDN1QDM0YzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL2czLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/d/b9a/wZwpmL4IDNwgzN2ADO2UzM1UTM1QDN5MjM5ADMwAjMwUzLwgzLyQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/c/54e/wZwpmLxIjM2cDO3UjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL1QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/9/4f7/wZwpmL2UDM2YjM3YzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLxUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/7/5c1/wZwpmLwIDOyAzM3YzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLwMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/4/f6b/wZwpmL1ETN0IDO5MjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLzYzLwYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/5/fa3/wZwpmL2ADOzEDMwUTMwEDN0UTMyITNykTO0EDMwAjMwUzL1EzL1gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/f/b79/wZwpmL3ATOzMDMxIDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL3UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/e/3bb/wZwpmLzUTNycjN2gzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4czL2MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
從中,可以看出,若(為質數)那么,,所以,.
如果我們要讓函式滿足:
![莫比烏斯反演](/img/1/e0e/wZwpmL4UTM1QjN0cTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3UzLzEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
那么通過以上推導,我們可以知道,所以我們作出以下猜測:
![莫比烏斯反演](/img/9/f2f/wZwpmL2QTOycjNygTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4UzLwIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
莫比烏斯反演定理
![莫比烏斯反演](/img/5/beb/wZwpmL3cjNycTN0ITMzEzM1UTM1QDN5MjM5ADMwAjMwUzLyEzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/c/16c/wZwpmL3MjM3cDNwIDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLyUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設 和 是定義在正整數集合上的兩個函式,定義如下。
![莫比烏斯反演](/img/5/beb/wZwpmL3cjNycTN0ITMzEzM1UTM1QDN5MjM5ADMwAjMwUzLyEzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
若函式滿足:
![莫比烏斯反演](/img/6/12f/wZwpmL0MDOzUDO3QjN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
則有
![莫比烏斯反演](/img/8/08e/wZwpmLwADM4YDOxMDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzAzLzczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
莫比烏斯反演定理證明
充分性證明:
![莫比烏斯反演](/img/c/520/wZwpmL0ETNxETO3QjN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzLwczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/3/5e0/wZwpmL2YjM2YzMzADN3UzM1UTM1QDN5MjM5ADMwAjMwUzLwQzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/9/e77/wZwpmLwEDOwETN4UDN3UzM1UTM1QDN5MjM5ADMwAjMwUzL1QzLzczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/3/1a1/wZwpmL4ADMyUTN2AzN2UzM1UTM1QDN5MjM5ADMwAjMwUzLwczL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
考慮到:
![莫比烏斯反演](/img/5/0bc/wZwpmLwcTO5QzNxQTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0kzL0EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
因此
![莫比烏斯反演](/img/5/59b/wZwpmLyczM5UjMwUzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL1czLyMzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/0/73e/wZwpmL4EDN4QzN4YTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL2kzL3UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
必要性證明:
![莫比烏斯反演](/img/6/fd6/wZwpmLwcDNzMjN0YzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL2czLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/b/6d1/wZwpmLyYDMyUjNwQDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0AzLwEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
考慮到:
![莫比烏斯反演](/img/7/a5d/wZwpmL3UzMwAjM3QzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL4UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/c/520/wZwpmL0ETNxETO3QjN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzLwczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
因此
![莫比烏斯反演](/img/6/fd6/wZwpmLwcDNzMjN0YzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL2czLwMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
莫比烏斯函式
![莫比烏斯反演](/img/d/5bf/wZwpmL0gjNwgzMyEjMwEDN0UTMyITNykTO0EDMwAjMwUzLxIzL0UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/c/d22/wZwpmL3MDOzEzM2YzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLyUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
定義當時,
![莫比烏斯反演](/img/0/33a/wZwpmLygTM5EjM3cTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3UzLxgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/5/fa3/wZwpmL2ADOzEDMwUTMwEDN0UTMyITNykTO0EDMwAjMwUzL1EzL1gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![莫比烏斯反演](/img/6/5c8/wZwpmLwgDOzYTNzQTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0kzLwczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
當(為不同的質數,且次數都為1),
![莫比烏斯反演](/img/4/71d/wZwpmL3UjN1kDNyQDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0AzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
其餘情況
![莫比烏斯反演](/img/0/588/wZwpmL3gDM3MDM4MDMwEDN0UTMyITNykTO0EDMwAjMwUzLzAzLzMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
注意, 函式也為積性函式。證明略。
莫比烏斯反演的性質
![莫比烏斯反演](/img/9/f2f/wZwpmL2QTOycjNygTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4UzLwIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
性質一(莫比烏斯反演公式):
性質二: μ( n)是積性函式
![莫比烏斯反演](/img/8/e73/wZwpmLxITM2kzM3kTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL4UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
性質三:設f是算術函式,它的和函式 是積性函式,那么 f 也是積性函式。