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¤U¤@­¶: ¥¿¦]¼Æ­Ó¼Æ¤Î¥¿¦]¼Æ©M ¤W¤@­¶: Arithmetic Function «e¤@­¶: Arithmetic Function

Multiplicative Arithmetic Functions

¨Ã¤£¬O©Ò¦³ªº arithmetic function ³£«Ü¦³½ì, ¨ì©³­n±´°Q­þ¨Ç arithmetic function ©O? ³o§¹¥þ¨M©w©ó©ó­n±´°Qªº¬O¦³Ãö­þ¨Ç¾ã¼Æªº¯S©Ê. ¦]¬°¦b¦¹§Ú­ÌµÛ­«©ó¾ã¼Æªº¤À¸Ñ©Ê½è, ©Ò¥H§Ú­Ì±´°Q©Ò¿×ªº multiplicative arithmetic function.

Definition 2.1.1   §Ú­ÌºÙ±q $ \mathbb {N}$ ¨ì $ \mathbb {C}$ ªº¨ç¼Æ¬° arithmetic function. ­Y f : $ \mathbb {N}$$ \to$$ \mathbb {C}$ ¬O¤@­Ó arithmetic function º¡¨¬¹ï¥ô·N a, b $ \in$ $ \mathbb {N}$ ¥B gcd(a, b) = 1 ¬Ò¦³ f (ab) = f (a)f (b), «hºÙ f ¬O¤@­Ó multiplicative arithmetic function.

­nª`·N·í¤@­Ó arithmetic function f ¬O multiplicative ®É, f (ab) = f (a)f (b) ¨Ã¤£¤@©w¦¨¥ß. ³o¬O­n¦b gcd(a, b) = 1 ®É¤~¥i¥H½T©w¬O¹ïªº. ¦pªG f ªº©Ê½è±j¨ì¹ï¥ô·N a, b $ \in$ $ \mathbb {N}$ ¬Ò¦³ f (ab) = f (a)f (b), ¨º»ò§Ú­ÌºÙ f ¬O completely multiplicative. ¥Ñ©ó completely multiplicative arithmetic function ªº±ø¥ó¸û±j, ¥B¨ÃµL¤Ó¦h³oÃþ¦³½ìªº¨ç¼Æ, ©Ò¥H³o¸Ì§Ú­Ì¥u±Mª`©ó multiplicative arithmetic function.

§Ú­Ì¥ý¨Ó¬Ý¤@­Ó multiplicative arithmetic function ªº¨Ò¤l.

Example 2.1.2   §Ú­Ì¦Ò¼{ Möbius $ \mu$-function, ¨ä©w¸q¬°

$\displaystyle \mu$(n) = $\displaystyle \left\{\vphantom{
\begin{array}{ll}
1, & \hbox{­Y $n=1$;} \\
...
...n=p_1\cdots p_r$, ¨ä¤¤ $p_1,\dots,p_r$ ¬°¬Û²§½è¼Æ.} \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{ll}
1, & \hbox{­Y $n=1$;} \\
0, & \hbox{­Y¦s¦b½...
... \hbox{­Y $n=p_1\cdots p_r$, ¨ä¤¤ $p_1,\dots,p_r$ ¬°¬Û²§½è¼Æ.} \\
\end{array}$

§Ú­Ì¨ÓÅçÃÒ $ \mu$ ½T¬° multiplicative. ¦Ò¼{ a, b $ \in$ $ \mathbb {N}$ ¥B gcd(a, b) = 1. ¤µ­Y a = 1 «h¥Ñ $ \mu$(a) = $ \mu$(1) = 1 ±o $ \mu$(ab) = $ \mu$(b) = $ \mu$(a)$ \mu$(b). ¦P²z­Y b = 1 ¤]±o $ \mu$(ab) = $ \mu$(a)$ \mu$(b). ©Ò¥H§Ú­Ì¶È­n¦Ò¼{ a > 1 ¥B b > 1 ªº±¡§Î. ¥Ñºâ¼Æ°ò¥»©w²z (Theorem 1.5.1) §Ú­Ì¥i¥H±N a, b ¤À§O¼g¦¨ a = p1n1 ... prnr ¥H¤Î b = q1m1 . qtmt ªº§Î¦¡¨ä¤¤ ni, mj ¬Ò¤j©ó 0 ¥B¥Ñ©ó a, b ¤¬½è©Ò¦³ªº½è¼Æ pi ©M qj ¬Ò¬Û²§. ¤µ­Y ni ©Î mj ¤¤¦³¤@­Ó¤j©ó 1, ¤£¥¢¤@¯ë©Ê´N°²³] n1$ \ge$2, «h¥Ñ p12| a ¥B p12| ab, ª¾ $ \mu$(a) = 0 ¥B $ \mu$(ab) = 0, ¬G±o $ \mu$(ab) = $ \mu$(a)$ \mu$(b). ³Ì«á§Ú­Ì¥u³Ñ¤U n1 = ... = nr = 1 ¥B m1 = ... = mt = 1 ªº±¡ªp. ¦¹®É¥Ñ©ó ab = p1 ... pr . q1 ... qt ¥B p1,..., pr, q1,..., qt ¬°¬Û²§½è¼Æ±o $ \mu$(ab) = (- 1)r + t. µM¦Ó $ \mu$(a) = (- 1)r ¥B $ \mu$(b) = (- 1)t, ¬G±oÃÒ $ \mu$(ab) = $ \mu$(a)$ \mu$(b). ¤]´N¬O»¡ $ \mu$ ¬O¤@­Ó multiplicative arithmetic function.

­nª`·N $ \mu$ ¨Ã«D completely multiplicative. §Ú­Ì¥i¥H±q a = b = p, ¨ä¤¤ p ¬°½è¼Æªº±¡§Î¬Ý¥X. ¦¹®É $ \mu$(a) = $ \mu$(b) = 1 ¦ý¬O $ \mu$(ab) = 0, ¬Gª¾ $ \mu$(ab)$ \ne$$ \mu$(a)$ \mu$(b). ­nª¾¹D§A­nÃÒ¤@­Ó arithmetic function f ¬O multiplicative ®É, §A¥²¶·¦Ò¼{©Ò¦³ªº±¡ªp, §Y¹ï©Ò¦³º¡¨¬ gcd(a, b) = 1 ªº¥¿¾ã¼Æ a, b ¬Ò­n²Å¦X f (ab) = f (a)f (b), ¦Ó¤£¯à¶È¥N­Ó¨Ò¤lÅçÃÒ. ¦ý·í§A­n»¡ f ¤£¬O multiplicative ®É, ¥u­n§ä¨ì¤@²Õ a, b $ \in$ $ \mathbb {N}$ ¥B gcd(a, b) = 1 ·|¨Ï±o f (ab)$ \ne$f (a)f (b) §Y¥i.

±µ¤U¨Ó§Ú­Ì¨Ó¬Ý multiplicative arithmetic function ªº°ò¥»©Ê½è.

Proposition 2.1.3   °²³] f ¬O¤@­Ó«D 0 ªº multiplicative arithmetic function. «h f (1) = 1, ¥B­Y¹ï¥ô·Nªº½è¼Æ p ¥H¤Î t $ \in$ $ \mathbb {N}$, ³£¥iª¾ f (pt) ªº­È «h¹ï¥ô·N n $ \in$ $ \mathbb {N}$, f (n) ¤§­È´N¥i¥H½T©w.

µý ©ú. ¦] f ¬O multiplicative ¥B gcd(1, 1) = 1, ¬Gª¾ f (1) = f (1)f (1) ±oª¾ f (1) = 1 ©Î f (1) = 0. ­Y f (1) = 0, «h¹ï¥ô·N n $ \in$ $ \mathbb {N}$, ¥Ñ©ó gcd(n, 1) = 1, ¥i±o f (n) = f (n)f (1) = 0. ¤]´N¬O»¡ f ¬O 0 ¨ç¼Æ, ¦¹©M f ¬O«D 0 §t¼Æ¤§°²³]¥Ù¬Þ, ¬Gª¾ f (1) = 1.

²{¹ï¥ô·N n $ \in$ $ \mathbb {N}$, ­Y n = 1, «h¥Ñ«eª¾ f (n) = f (1) = 1. ­Y n > 1, «h¥Ñºâ¼Æ°ò¥»©w²zª¾ n = p1n1 ... prnr, ¨ä¤¤ pi ¬°¬Û²§½è¼Æ¥B ni $ \in$ $ \mathbb {N}$. ¬G¥Ñ f ¬O multiplicative ¥B gcd(p1n1, p2n2 ... prnr) = 1 ª¾ f (n) = f (p1n1p2n2 ... prnr) = f (p1n1)f (p2n2 ... prnr). Ä~Äò¤U¥h¨Ï¥Î¼Æ¾ÇÂk¯Çªkª¾ f (n) = f (p1n1) ... f (prnr). ¬G¥Ñ°²³]¤wª¾³o¨Ç f (pini) ¤§­È§Ú­Ì¥i½T©w f (n) ¤§­È. $ \qedsymbol$

¨Ì Proposition 2.1.3 §Ú­Ìª¾¦pªG f ¬O multiplicative arithmetic function, ¨º»ò­Y¯à´x´¤©Ò¦³½è¼Æ p ¥H¤Î t $ \in$ $ \mathbb {N}$ ¤¤ f (pt) ¤§­È¨º»ò´N¥i¥H§¹¥þ¤F¸Ñ f ³o¤@­Ó¨ç¼Æ. ¤£¹L«eÃD¬O­n½T»{ f ¬O§_¬° multiplicative. ©³¤U§Ú­Ì·|µ¹¤@­Ó±`¥Î¨Ó½T»{¬O multiplicative ªº¤èªk. ³o­Ó¤èªk¤£¥u¥i¥H®³¨Ó½T»{ multiplicative arithmetic function ¦Ó¥B¥i¥HÀ°§U§Ú­Ì³Ð³y³\¦h multiplicative arithmetic function. ¤£¹L­º¥ý§Ú­Ì»Ý­n¤@­Ó¸É§U©w²z.

Lemma 2.1.4   °²³] a, b $ \in$ $ \mathbb {N}$ ¥B gcd(a, b) = 1. ­Y d ¬O ab ªº¥¿¦]¼Æ, «h¦s¦b°ß¤@ªº a ªº¥¿¦]¼Æ d1 ¥H¤Î b ªº¥¿¦]¼Æ d2 ¨Ï±o d = d1d2.

µý ©ú. ³o¤S¬O¤@­Ó¦s¦b¤Î°ß¤@ªº°ÝÃD. ¦s¦b´N¬O­nÃÒ¦s¦b d1| a ¥B d2| b ¨Ï±o d = d1d2, ¦Ó°ß¤@´N¬O­nÃÒº¡¨¬³o±ø¥óªº¼gªk¥u¦³¤@ºØ.

­º¥ýÃÒ©ú¦s¦b©Ê. µ¹©w d| ab, ­n¦p¦ó§ä¨ì d1| a ¥B d2| b ¨Ï±o d = d1d2 ©O? ¥Ñ©ó­n¨D d1d2 = d ¥H¤Î d1| a ©Ò¥H d1 ¥²¶·¬O a ©M d ªº¤½¦]¼Æ. «ä¦Ò¤@¤U, §Ú­Ì¥i¦Ò¼{¨ú d1 ¬° a, d ªº³Ì¤j¤½¦]¼Æ, ³o¼Ë¤@¨Ó d2 = d /d1 ·|¤ñ¸û¤p¤ñ¸û¥i¯à¾ã°£ b. ´NÅý§Ú­Ì¨ú d1 = gcd(a, d ) ¬Ý¬Ý¬O§_¥i¦æ. ¦¹®É¥O d2 = d /d1, §Ú­Ì½T¹ê¦³ d = d1d2 ¥B d1| a. ¥u³Ñ¤U­nÅçÃÒ¬O§_ d2| b. µM¦Ó d| ab ¬Gª¾ (d /d1)|(a/d1)b. ¤S¥Ñ d1 = gcd(a, d ) ª¾ gcd(a/d1, d /d1) = 1 (Corollary 1.2.3), ¬G¥Ñ Proposition 1.2.7(1) ª¾ d /d1| b, ¤]´N¬O»¡ d2| b.

±µ¤U¨ÓÃҰߤ@©Ê. µ¹©w d| ab °²³]¦s¦b d1, d1', d2, d2' $ \in$ $ \mathbb {N}$ ¤À§Oº¡¨¬ d = d1d2, d1| a ¥B d2| b ¥H¤Î d = d1'd2', d1'| a ¥B d2'| b, §Ú­Ì­nÃÒ©ú d1 = d1' ¥B d2 = d2'. ¥Ñ©ó d1d2 = d1'd2', §Ú­Ìª¾ d1| d1'd2'. ¤S¥Ñ©ó d1| a, d2'| b ¥H¤Î gcd(a, b) = 1, §Ú­Ìª¾ gcd(d1, d2') = 1. ©Ò¥H¦A§Q¥Î Proposition 1.2.7(1) ±oª¾ d1| d1'. ¦P²z¥iÃÒ d1'| d1 ¦A¥[¤W d1, d1' $ \in$ $ \mathbb {N}$ ¬Gª¾ d1 = d1', ¥B±o d2 = d2'. $ \qedsymbol$

¦b Lemma 2.1.4 ¦³Ãö©ó¦s¦b©ÊªºÃÒ©ú¤¤§Ú­Ìµo²{¨Ã¥¼¥Î¨ì gcd(a, b) = 1 ªº°²³], ¤]´N¬O»¡¨Ã¤£»Ý°²³] gcd(a, b) = 1, ¹ï¥ô·N ab ªº¥¿¦]¼Æ³£¥i¥H§ä¨ì d1| a, d2| b ¨Ï±o d = d1d2. ¤£¹L¦bÃÒ©ú°ß¤@©Ê®É, gcd(a, b) = 1 ªº°²³]´N»Ý­n¤F. ¤ñ¤è»¡¦Ò¼{ a = 6, b = 4 ©M d = 6 ªº±¡§Î, §Ú­Ì¥i¥H¨ú d1 = 6, d2 = 1 ©M d1' = 3, d2' = 2 ³£º¡¨¬­n¨D, ©Ò¥H°ß¤@©Ê¦b¦¹±¡ªp¨Ã¤£¦¨¥ß. ¥Ñ¦¹§Ú­Ì¤]¦A¦¸±j½Õ°ß¤@©Êµ´¤£¯à¥Î¦]¬° a ©M d ªº³Ì¤j¤½¦]¼Æ¬O°ß¤@ªºª¾ d1 ¬O°ß¤@ªº¦Ó±oÃҰߤ@©Ê. ³o¬O¦]¬°µL±q±oª¾¬°¦ó d1 «D±o¬O a, b ªº³Ì¤j¤½¦]¼Æ¤£¥i. ©Ò¥H¦bÃÒ©ú°ß¤@©Ê®É, ¤j®aÁÙ¬O­n«ö³¡´N¯Z¦a¥ý°²³]¦³¨âºØ¼gªk¦A¥h»¡©ú³o¨âºØ¼gªk¬O¤@¼Ë, ³o¼Ëªº¤èªk¨Ó³B²z¤ñ¸û¤£·|¥X¿ù.

¨Æ¹ê¤W Lemma 2.1.4 §i¶D§Ú­Ì·í gcd(a, b) = 1 ®É, ­Y d1,..., di,..., dr ©M e1,..., ej,..., es ¤À§O¬O a ©M b ©Ò¦³ªº¬Û²§¥¿¦]¼Æ, «h d1e1,..., diej,..., dres ·|¬O ab ©Ò¦³ªº¬Û²§¥¿¦]¼Æ. ³o¬O¦]¬°³o¨Ç diej ¤@©w¬O ab ªº¥¿¦]¼Æ, ¦A¥[¤W Lemma 2.1.4 §i¶D§Ú­Ì ab ªº¥¿¤½¦]¼Æ¤@©w¥i¥H¼g¦¨ diej ªº§Î¦¡¦Ó¥B³o¨Ç diej ¤@©w¬Û²§. ±µ¤U¨Ó§Ú­Ì´N¬O­n¥Î³o©Ê½è¨Ó§Q¥Î¤@­Ó¤wª¾ªº multiplicative arithmetic function ±o¨ì·sªº multiplicative arithmetic function.

Theorem 2.1.5   °²³] f : $ \mathbb {N}$$ \to$$ \mathbb {C}$ ¬O¤@­Ó multiplicative arithmetic function. ¦Ò¼{¨ç¼Æ F : $ \mathbb {N}$$ \to$$ \mathbb {C}$ ¨ä©w¸q¬°¹ï¥ô·N n $ \in$ $ \mathbb {N}$,

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

«h F ¬O¤@­Ó multiplicative arithmetic function.

µý ©ú. ­º¥ý¸ÑÄÀ¤@¤U F(n) = $ \sum_{d\vert n,d>0}^{}$f (d ) ³o²Å¸¹ªí¥Ü¦pªG d1,..., dr ¬O n ªº©Ò¦³¬Û²§¥¿¦]¼Æ¨º»ò F(n) = f (d1) + ... + f (dr). §Ú­Ì­nÃÒ©ú F ¬O multiplicative ´N¬O­nÃÒ©ú·í a, b $ \in$ $ \mathbb {N}$ ¥B gcd(a, b) = 1 ®É F(ab) = F(a)F(b).

²{°²³] d1,..., di,...dr ©M e1,..., ej,..., es ¤À§O¬O a ©M b ©Ò¦³ªº¥¿¦]¼Æ. §Ú­Ì¦³ F(a) = f (d1) + ... + f (di) + ... + f (dr) ¥H¤Î F(b) = f (e1) + ... + f (ej) + ... + f (es). ¦]¦¹ª¾ F(a)F(b) = f (d1)f (e1) + ... + f (di)f (ej) + ... + f (dr)f (es). ¥Ñ©ó gcd(a, b) = 1 ¦Ó di, ej ¤À§O¬O a, b ªº¦]¼Æ, §Ú­Ìª¾ gcd(di, ej) = 1. ¦A¥[¤W f ¬O multiplicative, ¬G±o¹ï©Ò¦³ di, ej ¬Ò¦³ f (di)f (ej) = f (diej). ¦]¦¹±o F(a)F(b) = f (d1e1) + ... + f (diej) + ... + f (dres). µM¦Ó Lemma 2.1.4 §i¶D§Ú­Ì¥Ñ©ó gcd(a, b) = 1, ³o¨Ç d1e1,..., diej,..., dres ­è¦n´N¬O ab ©Ò¦³ªº¬Û²§¥¿¦]¼Æ, ¬G±oÃÒ F(ab) = F(a)F(b). $ \qedsymbol$

³Ì«á§Ú­Ì¨Ó¬Ý¬Ý Example 2.1.2 ¤¤ªº $ \mu$ §Q¥Î Theorem 2.1.5 ©Ò³Ð³y¥X¨Óªº multiplicative arithmetic function ¬°¦ó.

Example 2.1.6   ¥O $ \delta$ : $ \mathbb {N}$$ \to$$ \mathbb {C}$ ¬O¤@­Ó arithmetic function ¨ä©w¸q¬°, ¹ï¥ô·N n $ \in$ $ \mathbb {N}$,

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

¨ä¤¤ $ \mu$ ¬O möbius $ \mu$-function. ¦]¬° $ \mu$ ¬O multiplicative, ¥Ñ Theorem 2.1.5 ª¾ $ \delta$ ¬O multiplicative. ¬G­n¨M©w $ \delta$ ¤§­È¥Ñ Proposition 2.1.3 ª¾¥u­n¥ý¦Ò¼{ $ \delta$(pt) ¤§­È§Y¥i, ¨ä¤¤ p ¬O½è¼Æ t $ \in$ $ \mathbb {N}$. µM¦Ó pt ©Ò¦³ªº¥¿¦]¼Æ¬° 1, p, p2,..., pt, ¬G¥Ñ©w¸qª¾

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

¬G­Y n > 1, «h¥Ñ n = p1n1 ... prnr ª¾ $ \delta$(n) = $ \delta$(p1n1) ... $ \delta$(prnr) = 0. µM¦Ó¥Ñ©w¸q $ \delta$(1) = $ \mu$(1) = 1, ©Ò¥H§Ú­Ì¥i±o

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


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¤U¤@­¶: ¥¿¦]¼Æ­Ó¼Æ¤Î¥¿¦]¼Æ©M ¤W¤@­¶: Arithmetic Function «e¤@­¶: Arithmetic Function
Li 2007-06-28