next up previous
�U�@��: ���]�ƭӼƤΥ��]�ƩM �W�@��: Arithmetic Function �e�@��: Arithmetic Function

Multiplicative Arithmetic Functions

�ä��O�Ҧ��� arithmetic function ���ܦ���, �쩳�n���Q���� arithmetic function �O? �o�����M�w���n���Q���O�������Ǿ�ƪ��S��. �]���b���ڭ̵ۭ����ƪ����ѩʽ�, �ҥH�ڭ̱��Q�ҿת� multiplicative arithmetic function.

Definition 2.1.1   �ڭ̺ٱq $ \mathbb {N}$ �� $ \mathbb {C}$ ����Ƭ� arithmetic function. �Y f : $ \mathbb {N}$$ \to$$ \mathbb {C}$ �O�@�� arithmetic function ��������N a, b $ \in$ $ \mathbb {N}$ �B gcd(a, b) = 1 �Ҧ� f (ab) = f (a)f (b), �h�� f �O�@�� multiplicative arithmetic function.

�n�`�N���@�� arithmetic function f �O multiplicative ��, f (ab) = f (a)f (b) �ä��@�w����. �o�O�n�b gcd(a, b) = 1 �ɤ~�i�H�T�w�O�諸. �p�G f ���ʽ�j�����N a, b $ \in$ $ \mathbb {N}$ �Ҧ� f (ab) = f (a)f (b), ����ڭ̺� f �O completely multiplicative. �ѩ� completely multiplicative arithmetic function ��������j, �B�õL�Ӧh�o�����쪺���, �ҥH�o�̧ڭ̥u�M�`�� multiplicative arithmetic function.

�ڭ̥��Ӭݤ@�� multiplicative arithmetic function ���Ҥl.

Example 2.1.2   �ڭ̦Ҽ{ Möbius $ \mu$-function, ��w�q��

$\displaystyle \mu$(n) = $\displaystyle \left\{\vphantom{
\begin{array}{ll}
1, & \hbox{�Y $n=1$;} \\
...
...n=p_1\cdots p_r$, �䤤 $p_1,\dots,p_r$ ���۲����.} \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{ll}
1, & \hbox{�Y $n=1$;} \\
0, & \hbox{�Y�s�b�...
... \hbox{�Y $n=p_1\cdots p_r$, �䤤 $p_1,\dots,p_r$ ���۲����.} \\
\end{array}$

�ڭ̨����� $ \mu$ �T�� multiplicative. �Ҽ{ a, b $ \in$ $ \mathbb {N}$ �B gcd(a, b) = 1. ���Y a = 1 �h�� $ \mu$(a) = $ \mu$(1) = 1 �o $ \mu$(ab) = $ \mu$(b) = $ \mu$(a)$ \mu$(b). �P�z�Y b = 1 �]�o $ \mu$(ab) = $ \mu$(a)$ \mu$(b). �ҥH�ڭ̶ȭn�Ҽ{ a > 1 �B b > 1 ������. �Ѻ�ư򥻩w�z (Theorem 1.5.1) �ڭ̥i�H�N a, b ���O�g�� a = p1n1 ... prnr �H�� b = q1m1 . qtmt ���Φ��䤤 ni, mj �Ҥj�� 0 �B�ѩ� a, b ����Ҧ������ pi �M qj �Ҭ۲�. ���Y ni �� mj �����@�Ӥj�� 1, �����@��ʴN���] n1$ \ge$2, �h�� p12| a �B p12| ab, �� $ \mu$(a) = 0 �B $ \mu$(ab) = 0, �G�o $ \mu$(ab) = $ \mu$(a)$ \mu$(b). �̫�ڭ̥u�ѤU n1 = ... = nr = 1 �B m1 = ... = mt = 1 �����p. ���ɥѩ� ab = p1 ... pr . q1 ... qt �B p1,..., pr, q1,..., qt ���۲���Ʊo $ \mu$(ab) = (- 1)r + t. �M�� $ \mu$(a) = (- 1)r �B $ \mu$(b) = (- 1)t, �G�o�� $ \mu$(ab) = $ \mu$(a)$ \mu$(b). �]�N�O�� $ \mu$ �O�@�� multiplicative arithmetic function.

�n�`�N $ \mu$ �ëD completely multiplicative. �ڭ̥i�H�q a = b = p, �䤤 p ����ƪ����άݥX. ���� $ \mu$(a) = $ \mu$(b) = 1 ���O $ \mu$(ab) = 0, �G�� $ \mu$(ab)$ \ne$$ \mu$(a)$ \mu$(b). �n���D�A�n�Ҥ@�� arithmetic function f �O multiplicative ��, �A�����Ҽ{�Ҧ������p, �Y��Ҧ����� gcd(a, b) = 1 ������� a, b �ҭn�ŦX f (ab) = f (a)f (b), �Ӥ���ȥN�ӨҤl����. �����A�n�� f ���O multiplicative ��, �u�n���@�� a, b $ \in$ $ \mathbb {N}$ �B gcd(a, b) = 1 �|�ϱo f (ab)$ \ne$f (a)f (b) �Y�i.

���U�ӧڭ̨Ӭ� multiplicative arithmetic function ���򥻩ʽ�.

Proposition 2.1.3   ���] f �O�@�ӫD 0 �� multiplicative arithmetic function. �h f (1) = 1, �B�Y����N����� p �H�� t $ \in$ $ \mathbb {N}$, ���i�� f (pt) ���� �h����N n $ \in$ $ \mathbb {N}$, f (n) ���ȴN�i�H�T�w.

�� ��. �] f �O multiplicative �B gcd(1, 1) = 1, �G�� f (1) = f (1)f (1) �o�� f (1) = 1 �� f (1) = 0. �Y f (1) = 0, �h����N n $ \in$ $ \mathbb {N}$, �ѩ� gcd(n, 1) = 1, �i�o f (n) = f (n)f (1) = 0. �]�N�O�� f �O 0 ���, ���M f �O�D 0 �t�Ƥ����]�٬�, �G�� f (1) = 1.

�{����N n $ \in$ $ \mathbb {N}$, �Y n = 1, �h�ѫe�� f (n) = f (1) = 1. �Y n > 1, �h�Ѻ�ư򥻩w�z�� n = p1n1 ... prnr, �䤤 pi ���۲���ƥB ni $ \in$ $ \mathbb {N}$. �G�� f �O multiplicative �B gcd(p1n1, p2n2 ... prnr) = 1 �� f (n) = f (p1n1p2n2 ... prnr) = f (p1n1)f (p2n2 ... prnr). �~��U�h�ϥμƾ��k�Ǫk�� f (n) = f (p1n1) ... f (prnr). �G�Ѱ��]�w���o�� f (pini) ���ȧڭ̥i�T�w f (n) ����. $ \qedsymbol$

�� Proposition 2.1.3 �ڭ̪��p�G f �O multiplicative arithmetic function, ����Y��x���Ҧ���� p �H�� t $ \in$ $ \mathbb {N}$ �� f (pt) ���Ȩ���N�i�H�����F�� f �o�@�Ө��. ���L�e�D�O�n�T�{ f �O�_�� multiplicative. ���U�ڭ̷|���@�ӱ`�ΨӽT�{�O multiplicative ����k. �o�Ӥ�k���u�i�H���ӽT�{ multiplicative arithmetic function �ӥB�i�H���U�ڭ̳гy�\�h multiplicative arithmetic function. ���L�����ڭ̻ݭn�@�ӸɧU�w�z.

Lemma 2.1.4   ���] a, b $ \in$ $ \mathbb {N}$ �B gcd(a, b) = 1. �Y d �O ab �����]��, �h�s�b�ߤ@�� a �����]�� d1 �H�� b �����]�� d2 �ϱo d = d1d2.

�� ��. �o�S�O�@�Ӧs�b�ΰߤ@�����D. �s�b�N�O�n�Ҧs�b d1| a �B d2| b �ϱo d = d1d2, �Ӱߤ@�N�O�n�Һ����o���󪺼g�k�u���@��.

�����ҩ��s�b��. ���w d| ab, �n�p���� d1| a �B d2| b �ϱo d = d1d2 �O? �ѩ�n�D d1d2 = d �H�� d1| a �ҥH d1 �����O a �M d �����]��. ��Ҥ@�U, �ڭ̥i�Ҽ{�� d1 �� a, d ���̤j���]��, �o�ˤ@�� d2 = d /d1 �|����p����i��㰣 b. �N���ڭ̨� d1 = gcd(a, d ) �ݬݬO�_�i��. ���ɥO d2 = d /d1, �ڭ̽T�꦳ d = d1d2 �B d1| a. �u�ѤU�n���ҬO�_ d2| b. �M�� d| ab �G�� (d /d1)|(a/d1)b. �S�� d1 = gcd(a, d ) �� gcd(a/d1, d /d1) = 1 (Corollary 1.2.3), �G�� Proposition 1.2.7(1) �� d /d1| b, �]�N�O�� d2| b.

���U���Ұߤ@��. ���w d| ab ���]�s�b d1, d1', d2, d2' $ \in$ $ \mathbb {N}$ ���O���� d = d1d2, d1| a �B d2| b �H�� d = d1'd2', d1'| a �B d2'| b, �ڭ̭n�ҩ� d1 = d1' �B d2 = d2'. �ѩ� d1d2 = d1'd2', �ڭ̪� d1| d1'd2'. �S�ѩ� d1| a, d2'| b �H�� gcd(a, b) = 1, �ڭ̪� gcd(d1, d2') = 1. �ҥH�A�Q�� Proposition 1.2.7(1) �o�� d1| d1'. �P�z�i�� d1'| d1 �A�[�W d1, d1' $ \in$ $ \mathbb {N}$ �G�� d1 = d1', �B�o d2 = d2'. $ \qedsymbol$

�b Lemma 2.1.4 ������s�b�ʪ��ҩ����ڭ̵o�{�å��Ψ� gcd(a, b) = 1 �����], �]�N�O���ä��ݰ��] gcd(a, b) = 1, ����N ab �����]�Ƴ��i�H��� d1| a, d2| b �ϱo d = d1d2. ���L�b�ҩ��ߤ@�ʮ�, gcd(a, b) = 1 �����]�N�ݭn�F. ��軡�Ҽ{ a = 6, b = 4 �M d = 6 ������, �ڭ̥i�H�� d1 = 6, d2 = 1 �M d1' = 3, d2' = 2 �������n�D, �ҥH�ߤ@�ʦb�����p�ä�����. �Ѧ��ڭ̤]�A���j�հߤ@�ʵ�����Φ]�� a �M d ���̤j���]�ƬO�ߤ@���� d1 �O�ߤ@���ӱo�Ұߤ@��. �o�O�]���L�q�o������ d1 �D�o�O a, b ���̤j���]�Ƥ��i. �ҥH�b�ҩ��ߤ@�ʮ�, �j�a�٬O�n�����N�Z�a�����]����ؼg�k�A�h�����o��ؼg�k�O�@��, �o�˪���k�ӳB�z������|�X��.

�ƹ�W Lemma 2.1.4 �i�D�ڭ̷� gcd(a, b) = 1 ��, �Y d1,..., di,..., dr �M e1,..., ej,..., es ���O�O a �M b �Ҧ����۲����]��, �h d1e1,..., diej,..., dres �|�O ab �Ҧ����۲����]��. �o�O�]���o�� diej �@�w�O ab �����]��, �A�[�W Lemma 2.1.4 �i�D�ڭ� ab �������]�Ƥ@�w�i�H�g�� diej ���Φ��ӥB�o�� diej �@�w�۲�. ���U�ӧڭ̴N�O�n�γo�ʽ�ӧQ�Τ@�Ӥw���� multiplicative arithmetic function �o��s�� multiplicative arithmetic function.

Theorem 2.1.5   ���] f : $ \mathbb {N}$$ \to$$ \mathbb {C}$ �O�@�� multiplicative arithmetic function. �Ҽ{��� F : $ \mathbb {N}$$ \to$$ \mathbb {C}$ ��w�q������N n $ \in$ $ \mathbb {N}$,

F(n) = $\displaystyle \sum_{d\vert n,d>0}^{}$f (d ),

�h F �O�@�� multiplicative arithmetic function.

�� ��. ���������@�U F(n) = $ \sum_{d\vert n,d>0}^{}$f (d ) �o�Ÿ����ܦp�G d1,..., dr �O n ���Ҧ��۲����]�ƨ��� F(n) = f (d1) + ... + f (dr). �ڭ̭n�ҩ� F �O multiplicative �N�O�n�ҩ��� a, b $ \in$ $ \mathbb {N}$ �B gcd(a, b) = 1 �� F(ab) = F(a)F(b).

�{���] d1,..., di,...dr �M e1,..., ej,..., es ���O�O a �M b �Ҧ������]��. �ڭ̦� F(a) = f (d1) + ... + f (di) + ... + f (dr) �H�� F(b) = f (e1) + ... + f (ej) + ... + f (es). �]���� F(a)F(b) = f (d1)f (e1) + ... + f (di)f (ej) + ... + f (dr)f (es). �ѩ� gcd(a, b) = 1 �� di, ej ���O�O a, b ���]��, �ڭ̪� gcd(di, ej) = 1. �A�[�W f �O multiplicative, �G�o��Ҧ� di, ej �Ҧ� f (di)f (ej) = f (diej). �]���o F(a)F(b) = f (d1e1) + ... + f (diej) + ... + f (dres). �M�� Lemma 2.1.4 �i�D�ڭ̥ѩ� gcd(a, b) = 1, �o�� d1e1,..., diej,..., dres ��n�N�O ab �Ҧ����۲����]��, �G�o�� F(ab) = F(a)F(b). $ \qedsymbol$

�̫�ڭ̨Ӭݬ� Example 2.1.2 ���� $ \mu$ �Q�� Theorem 2.1.5 �ҳгy�X�Ӫ� multiplicative arithmetic function ����.

Example 2.1.6   �O $ \delta$ : $ \mathbb {N}$$ \to$$ \mathbb {C}$ �O�@�� arithmetic function ��w�q��, ����N n $ \in$ $ \mathbb {N}$,

$\displaystyle \delta$(n) = $\displaystyle \sum_{d\vert n,d>0}^{}$$\displaystyle \mu$(d ),

�䤤 $ \mu$ �O möbius $ \mu$-function. �]�� $ \mu$ �O multiplicative, �� Theorem 2.1.5 �� $ \delta$ �O multiplicative. �G�n�M�w $ \delta$ ���ȥ� Proposition 2.1.3 ���u�n���Ҽ{ $ \delta$(pt) ���ȧY�i, �䤤 p �O��� t $ \in$ $ \mathbb {N}$. �M�� pt �Ҧ������]�Ƭ� 1, p, p2,..., pt, �G�ѩw�q��

$\displaystyle \delta$(pt) = $\displaystyle \mu$(1) + $\displaystyle \mu$(p) + $\displaystyle \mu$(p2) + ... $\displaystyle \mu$(pt) = 1 - 1 + 0 + ... + 0 = 0.

�G�Y n > 1, �h�� n = p1n1 ... prnr �� $ \delta$(n) = $ \delta$(p1n1) ... $ \delta$(prnr) = 0. �M�ӥѩw�q $ \delta$(1) = $ \mu$(1) = 1, �ҥH�ڭ̥i�o

$\displaystyle \delta$(n) = $\displaystyle \sum_{d\vert n,d>0}^{}$$\displaystyle \mu$(d )= $\displaystyle \left\{\vphantom{
\begin{array}{ll}
1, & \hbox{�� $n=1$;} \\
0, & \hbox{�� $n>1$.} \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{ll}
1, & \hbox{�� $n=1$;} \\
0, & \hbox{�� $n>1$.} \\
\end{array}$


next up previous
�U�@��: ���]�ƭӼƤΥ��]�ƩM �W�@��: Arithmetic Function �e�@��: Arithmetic Function
Li 2007-06-28