�n�`�N���@�� arithmetic function f �O multiplicative ��,
f (ab) = f (a)f (b) ��@�w����. �o�O�n�b
gcd(a, b) = 1
�ɤ~�i�H�T�w�O�諸. �p�G f ���ʽ�j�����N
a, b
�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.
�n�`�N �ëD completely multiplicative. �ڭ̥i�H�q a = b = p,
�䤤 p ����ƪ����άݥX. ����
(a) =
(b) = 1 ���O
(ab) = 0,
�G��
(ab)
(a)
(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
�B
gcd(a, b) = 1 �|�ϱo
f (ab)
f (a)f (b) �Y�i.
���U�ӧڭ̨Ӭ� multiplicative arithmetic function ���ʽ�.
�{����N
n
, �Y n = 1, �h�ѫe��
f (n) = f (1) = 1. �Y n > 1,
�h�Ѻ�ưw�z��
n = p1n1 ... prnr, �䤤 pi
���۲���ƥB
ni
. �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) ����.
�� Proposition 2.1.3 �ڭ̪��p�G f �O multiplicative arithmetic
function, ����Y��x���Ҧ���� p �H��
t
�� 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.
�����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'
���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'
�G�� d1 = d1', �B�o d2 = d2'.
�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.
�{���]
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).
�̫�ڭ̨Ӭݬ� Example 2.1.2 ���� �Q�� Theorem
2.1.5 �ҳгy�X�Ӫ� multiplicative arithmetic function ����.