1703:
1208:
1698:{\displaystyle {\begin{aligned}f\cdot \left(g*h\right)(n)&=f(n)\cdot \sum _{d|n}g(d)h\left({\frac {n}{d}}\right)=\\&=\sum _{d|n}f(n)\cdot (g(d)h\left({\frac {n}{d}}\right))=\\&=\sum _{d|n}(f(d)f\left({\frac {n}{d}}\right))\cdot (g(d)h\left({\frac {n}{d}}\right)){\text{ (since }}f{\text{ is completely multiplicative) }}=\\&=\sum _{d|n}(f(d)g(d))\cdot (f\left({\frac {n}{d}}\right)h\left({\frac {n}{d}}\right))\\&=(f\cdot g)*(f\cdot h).\end{aligned}}}
681:
877:
of two multiplicative functions is multiplicative, the
Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative. Arithmetic functions which can be written as the Dirichlet convolution of two completely multiplicative functions are said to be quadratics or
373:
1910:
241:
676:{\displaystyle {\begin{aligned}f(1)=f(1\cdot 1)&\iff f(1)=f(1)f(1)\\&\iff f(1)=f(1)^{2}\\&\iff f(1)^{2}-f(1)=0\\&\iff f(1)\left(f(1)-1\right)=0\\&\iff f(1)=0\lor f(1)=1.\end{aligned}}}
1754:
1213:
378:
1020:
731:
1098:
913:
54:. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.
1167:
1050:
299:
150:
1191:
1132:
1746:
933:
368:
339:
319:
2021:
V. Laohakosol, Logarithmic operators and characterizations of completely multiplicative functions, Southeast Asian Bull. Math. 25 (2001) no. 2, 273–281.
2016:
E. Langford, Distributivity over the
Dirichlet product and completely multiplicative arithmetical functions, Amer. Math. Monthly 80 (1973) 411–414.
878:
specially multiplicative multiplicative functions. They are rational arithmetic functions of order (2, 0) and obey the Busche-Ramanujan identity.
2011:
P. Haukkanen, On characterizations of completely multiplicative arithmetical functions, in Number theory, Turku, de
Gruyter, 2001, pp. 115–123.
881:
There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function
155:
17:
1905:{\displaystyle L(s,a)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}=\prod _{p}{\biggl (}1-{\frac {a(p)}{p^{s}}}{\biggr )}^{-1},}
2006:
T. M. Apostol, Some properties of completely multiplicative arithmetical functions, Amer. Math. Monthly 78 (1971) 266-271.
951:
827:
A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the
2039:
1975:
828:
2026:
K. L. Yocom, Totally multiplicative functions in regular convolution rings, Canad. Math. Bull. 16 (1973) 119–128.
695:
685:
The definition above can be rephrased using the language of algebra: A completely multiplicative function is a
1915:
which means that the sum all over the natural numbers is equal to the product all over the prime numbers.
1170:
1053:
1065:
1939:
1714:
892:
51:
1967:
1960:
874:
1137:
1035:
269:
120:
1929:
1176:
1105:
67:
1722:
918:
344:
8:
1924:
808:
63:
804:
324:
304:
936:
1971:
1029:
886:
1934:
1717:
1194:
35:
816:
62:
A completely multiplicative function (or totally multiplicative function) is an
71:
1056:. One consequence of this is that for any completely multiplicative function
2033:
812:
31:
733:(that is, the positive integers under multiplication) to some other monoid.
686:
885:
is multiplicative then it is completely multiplicative if and only if its
942:
Completely multiplicative functions also satisfy a distributive law. If
807:
is a non-trivial example of a completely multiplicative function as are
46:. A weaker condition is also important, respecting only products of
742:
745:
with leading coefficient 1: For any particular positive integer
741:
The easiest example of a completely multiplicative function is a
47:
236:{\displaystyle \forall a,b\in {\text{domain}}(f),f(ab)=f(a)f(b)}
690:
266:, so this is not a very strong restriction. If one did not fix
1757:
1725:
1211:
1179:
1140:
1108:
1068:
1038:
954:
921:
895:
698:
376:
347:
327:
307:
272:
158:
123:
1102:
which can be deduced from the above by putting both
38:
which respect products are important and are called
1959:
1904:
1740:
1697:
1185:
1161:
1126:
1092:
1044:
1015:{\displaystyle f\cdot (g*h)=(f\cdot g)*(f\cdot h)}
1014:
927:
907:
725:
675:
362:
333:
313:
293:
235:
144:
1885:
1843:
1200:
2031:
835:is a product of powers of distinct primes, say
1713:The L-function of completely (or totally)
629:
625:
571:
567:
519:
515:
473:
469:
422:
418:
1530: is completely multiplicative)
726:{\displaystyle (\mathbb {Z} ^{+},\cdot )}
704:
1957:
50:numbers, and such functions are called
14:
2032:
1962:Introduction to Analytic Number Theory
1708:
341:are possibilities for the value of
40:completely multiplicative functions
24:
1795:
946:is completely multiplicative then
159:
25:
2051:
829:fundamental theorem of arithmetic
1093:{\displaystyle f*f=\tau \cdot f}
262:) = 0 for all positive integers
44:totally multiplicative functions
1993:
1984:
1951:
1866:
1860:
1812:
1806:
1773:
1761:
1735:
1729:
1685:
1673:
1667:
1655:
1642:
1597:
1591:
1588:
1582:
1576:
1570:
1564:
1555:
1517:
1493:
1487:
1481:
1475:
1451:
1445:
1439:
1430:
1405:
1381:
1375:
1369:
1363:
1357:
1345:
1299:
1293:
1281:
1266:
1260:
1247:
1241:
1201:Proof of distributive property
1150:
1144:
1009:
997:
991:
979:
973:
961:
720:
699:
660:
654:
639:
633:
626:
598:
592:
581:
575:
568:
551:
545:
530:
523:
516:
499:
492:
483:
477:
470:
459:
453:
447:
441:
432:
426:
419:
411:
399:
390:
384:
357:
351:
282:
276:
250:(1) = 1, one could still have
230:
224:
218:
212:
203:
194:
185:
179:
133:
127:
13:
1:
1945:
822:
246:Without the requirement that
57:
7:
1918:
908:{\displaystyle \mu \cdot f}
736:
66:(that is, a function whose
10:
2056:
2040:Multiplicative functions
1054:pointwise multiplication
301:, one can see that both
52:multiplicative functions
1940:Multiplicative function
1906:
1799:
1742:
1699:
1187:
1163:
1162:{\displaystyle 1(n)=1}
1128:
1094:
1046:
1045:{\displaystyle \cdot }
1016:
929:
909:
727:
677:
370:in the following way:
364:
335:
315:
295:
294:{\displaystyle f(1)=1}
237:
146:
145:{\displaystyle f(1)=1}
18:Totally multiplicative
1966:. Springer. pp.
1958:Apostol, Tom (1976).
1907:
1779:
1743:
1700:
1188:
1186:{\displaystyle \tau }
1164:
1129:
1127:{\displaystyle g=h=1}
1095:
1047:
1017:
930:
910:
875:Dirichlet convolution
728:
678:
365:
336:
316:
296:
238:
147:
1930:Dirichlet L-function
1755:
1741:{\displaystyle a(n)}
1723:
1209:
1177:
1138:
1106:
1066:
1036:
952:
928:{\displaystyle \mu }
919:
893:
809:Dirichlet characters
696:
374:
363:{\displaystyle f(1)}
345:
325:
305:
270:
156:
121:
1925:Arithmetic function
117:In logic notation:
64:arithmetic function
27:Arithmetic function
1902:
1840:
1738:
1695:
1693:
1563:
1522: (since
1438:
1353:
1289:
1183:
1159:
1124:
1090:
1042:
1012:
925:
905:
805:Liouville function
723:
673:
671:
360:
331:
311:
291:
254:(1) = 0, but then
233:
142:
106:positive integers
1880:
1831:
1826:
1636:
1615:
1546:
1531:
1523:
1511:
1469:
1421:
1399:
1336:
1317:
1272:
1171:constant function
1030:Dirichlet product
887:Dirichlet inverse
334:{\displaystyle 1}
314:{\displaystyle 0}
177:
36:positive integers
16:(Redirected from
2047:
2000:
1997:
1991:
1988:
1982:
1981:
1965:
1955:
1935:Dirichlet series
1911:
1909:
1908:
1903:
1898:
1897:
1889:
1888:
1881:
1879:
1878:
1869:
1855:
1847:
1846:
1839:
1827:
1825:
1824:
1815:
1801:
1798:
1793:
1747:
1745:
1744:
1739:
1718:Dirichlet series
1709:Dirichlet series
1704:
1702:
1701:
1696:
1694:
1648:
1641:
1637:
1629:
1620:
1616:
1608:
1562:
1558:
1539:
1532:
1529:
1524:
1521:
1516:
1512:
1504:
1474:
1470:
1462:
1437:
1433:
1414:
1404:
1400:
1392:
1352:
1348:
1329:
1322:
1318:
1310:
1288:
1284:
1240:
1236:
1195:divisor function
1192:
1190:
1189:
1184:
1168:
1166:
1165:
1160:
1133:
1131:
1130:
1125:
1099:
1097:
1096:
1091:
1051:
1049:
1048:
1043:
1021:
1019:
1018:
1013:
934:
932:
931:
926:
914:
912:
911:
906:
732:
730:
729:
724:
713:
712:
707:
682:
680:
679:
674:
672:
621:
611:
607:
563:
538:
537:
511:
507:
506:
465:
369:
367:
366:
361:
340:
338:
337:
332:
320:
318:
317:
312:
300:
298:
297:
292:
242:
240:
239:
234:
178:
175:
151:
149:
148:
143:
21:
2055:
2054:
2050:
2049:
2048:
2046:
2045:
2044:
2030:
2029:
2003:
1999:Apostol pg. 49
1998:
1994:
1989:
1985:
1978:
1956:
1952:
1948:
1921:
1890:
1884:
1883:
1882:
1874:
1870:
1856:
1854:
1842:
1841:
1835:
1820:
1816:
1802:
1800:
1794:
1783:
1756:
1753:
1752:
1724:
1721:
1720:
1711:
1692:
1691:
1646:
1645:
1628:
1624:
1607:
1603:
1554:
1550:
1537:
1536:
1528:
1520:
1503:
1499:
1461:
1457:
1429:
1425:
1412:
1411:
1391:
1387:
1344:
1340:
1327:
1326:
1309:
1305:
1280:
1276:
1250:
1226:
1222:
1212:
1210:
1207:
1206:
1203:
1178:
1175:
1174:
1139:
1136:
1135:
1107:
1104:
1103:
1067:
1064:
1063:
1037:
1034:
1033:
1028:represents the
953:
950:
949:
937:Möbius function
920:
917:
916:
894:
891:
890:
825:
817:Legendre symbol
739:
708:
703:
702:
697:
694:
693:
670:
669:
619:
618:
588:
584:
561:
560:
533:
529:
509:
508:
502:
498:
463:
462:
414:
377:
375:
372:
371:
346:
343:
342:
326:
323:
322:
306:
303:
302:
271:
268:
267:
174:
157:
154:
153:
122:
119:
118:
72:natural numbers
60:
34:, functions of
28:
23:
22:
15:
12:
11:
5:
2053:
2043:
2042:
2028:
2027:
2023:
2022:
2018:
2017:
2013:
2012:
2008:
2007:
2002:
2001:
1992:
1990:Apostol, p. 36
1983:
1976:
1949:
1947:
1944:
1943:
1942:
1937:
1932:
1927:
1920:
1917:
1913:
1912:
1901:
1896:
1893:
1887:
1877:
1873:
1868:
1865:
1862:
1859:
1853:
1850:
1845:
1838:
1834:
1830:
1823:
1819:
1814:
1811:
1808:
1805:
1797:
1792:
1789:
1786:
1782:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1737:
1734:
1731:
1728:
1715:multiplicative
1710:
1707:
1706:
1705:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1649:
1647:
1644:
1640:
1635:
1632:
1627:
1623:
1619:
1614:
1611:
1606:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1561:
1557:
1553:
1549:
1545:
1542:
1540:
1538:
1535:
1527:
1519:
1515:
1510:
1507:
1502:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1473:
1468:
1465:
1460:
1456:
1453:
1450:
1447:
1444:
1441:
1436:
1432:
1428:
1424:
1420:
1417:
1415:
1413:
1410:
1407:
1403:
1398:
1395:
1390:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1351:
1347:
1343:
1339:
1335:
1332:
1330:
1328:
1325:
1321:
1316:
1313:
1308:
1304:
1301:
1298:
1295:
1292:
1287:
1283:
1279:
1275:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1251:
1249:
1246:
1243:
1239:
1235:
1232:
1229:
1225:
1221:
1218:
1215:
1214:
1202:
1199:
1182:
1158:
1155:
1152:
1149:
1146:
1143:
1123:
1120:
1117:
1114:
1111:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1041:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
924:
904:
901:
898:
824:
821:
738:
735:
722:
719:
716:
711:
706:
701:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
628:
624:
622:
620:
617:
614:
610:
606:
603:
600:
597:
594:
591:
587:
583:
580:
577:
574:
570:
566:
564:
562:
559:
556:
553:
550:
547:
544:
541:
536:
532:
528:
525:
522:
518:
514:
512:
510:
505:
501:
497:
494:
491:
488:
485:
482:
479:
476:
472:
468:
466:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
421:
417:
415:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
379:
359:
356:
353:
350:
330:
310:
290:
287:
284:
281:
278:
275:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
173:
170:
167:
164:
161:
141:
138:
135:
132:
129:
126:
59:
56:
26:
9:
6:
4:
3:
2:
2052:
2041:
2038:
2037:
2035:
2025:
2024:
2020:
2019:
2015:
2014:
2010:
2009:
2005:
2004:
1996:
1987:
1979:
1977:0-387-90163-9
1973:
1969:
1964:
1963:
1954:
1950:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1922:
1916:
1899:
1894:
1891:
1875:
1871:
1863:
1857:
1851:
1848:
1836:
1832:
1828:
1821:
1817:
1809:
1803:
1790:
1787:
1784:
1780:
1776:
1770:
1767:
1764:
1758:
1751:
1750:
1749:
1732:
1726:
1719:
1716:
1688:
1682:
1679:
1676:
1670:
1664:
1661:
1658:
1652:
1650:
1638:
1633:
1630:
1625:
1621:
1617:
1612:
1609:
1604:
1600:
1594:
1585:
1579:
1573:
1567:
1559:
1551:
1547:
1543:
1541:
1533:
1525:
1513:
1508:
1505:
1500:
1496:
1490:
1484:
1478:
1471:
1466:
1463:
1458:
1454:
1448:
1442:
1434:
1426:
1422:
1418:
1416:
1408:
1401:
1396:
1393:
1388:
1384:
1378:
1372:
1366:
1360:
1354:
1349:
1341:
1337:
1333:
1331:
1323:
1319:
1314:
1311:
1306:
1302:
1296:
1290:
1285:
1277:
1273:
1269:
1263:
1257:
1254:
1252:
1244:
1237:
1233:
1230:
1227:
1223:
1219:
1216:
1205:
1204:
1198:
1196:
1180:
1172:
1156:
1153:
1147:
1141:
1121:
1118:
1115:
1112:
1109:
1100:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1061:
1059:
1055:
1039:
1031:
1027:
1022:
1006:
1003:
1000:
994:
988:
985:
982:
976:
970:
967:
964:
958:
955:
947:
945:
940:
938:
922:
902:
899:
896:
888:
884:
879:
876:
871:
869:
865:
861:
857:
853:
849:
845:
842:
838:
834:
830:
820:
818:
814:
813:Jacobi symbol
810:
806:
801:
800:(1) = 1 = 1.
799:
795:
791:
787:
783:
779:
776:
772:
768:
764:
760:
756:
752:
748:
744:
734:
717:
714:
709:
692:
688:
683:
666:
663:
657:
651:
648:
645:
642:
636:
630:
623:
615:
612:
608:
604:
601:
595:
589:
585:
578:
572:
565:
557:
554:
548:
542:
539:
534:
526:
520:
513:
503:
495:
489:
486:
480:
474:
467:
456:
450:
444:
438:
435:
429:
423:
416:
408:
405:
402:
396:
393:
387:
381:
354:
348:
328:
308:
288:
285:
279:
273:
265:
261:
257:
253:
249:
244:
227:
221:
215:
209:
206:
200:
197:
191:
188:
182:
171:
168:
165:
162:
139:
136:
130:
124:
115:
113:
109:
105:
101:
97:
93:
89:
85:
81:
77:
74:), such that
73:
69:
65:
55:
53:
49:
45:
41:
37:
33:
32:number theory
19:
1995:
1986:
1961:
1953:
1914:
1712:
1101:
1062:
1057:
1025:
1023:
948:
943:
941:
882:
880:
872:
867:
863:
859:
855:
851:
847:
843:
840:
836:
832:
826:
802:
797:
793:
789:
785:
781:
777:
774:
770:
766:
762:
758:
754:
750:
746:
740:
687:homomorphism
684:
263:
259:
255:
251:
247:
245:
116:
111:
107:
103:
99:
95:
91:
87:
83:
79:
78:(1) = 1 and
75:
61:
43:
39:
29:
1052:represents
831:. Thus, if
1946:References
1748:satisfies
873:While the
846:..., then
823:Properties
58:Definition
1892:−
1852:−
1833:∏
1796:∞
1781:∑
1680:⋅
1671:∗
1662:⋅
1595:⋅
1548:∑
1479:⋅
1423:∑
1367:⋅
1338:∑
1274:∑
1270:⋅
1231:∗
1220:⋅
1181:τ
1085:⋅
1082:τ
1073:∗
1040:⋅
1004:⋅
995:∗
986:⋅
968:∗
959:⋅
923:μ
900:⋅
897:μ
749:, define
718:⋅
689:from the
649:∨
627:⟺
602:−
569:⟺
540:−
517:⟺
471:⟺
420:⟺
406:⋅
172:∈
160:∀
2034:Category
1919:See also
1134:, where
1060:one has
815:and the
743:monomial
737:Examples
102:) holds
1193:is the
1173:. Here
1169:is the
935:is the
796:), and
761:. Then
104:for all
70:is the
48:coprime
1974:
1024:where
915:where
870:) ...
811:, the
691:monoid
176:domain
68:domain
1032:and
769:) = (
1972:ISBN
854:) =
803:The
773:) =
757:) =
321:and
152:and
110:and
86:) =
889:is
42:or
30:In
2036::
1970:.
1968:30
1197:.
939:.
862:)
839:=
819:.
780:=
771:bc
767:bc
667:1.
243:.
114:.
84:ab
1980:.
1900:,
1895:1
1886:)
1876:s
1872:p
1867:)
1864:p
1861:(
1858:a
1849:1
1844:(
1837:p
1829:=
1822:s
1818:n
1813:)
1810:n
1807:(
1804:a
1791:1
1788:=
1785:n
1777:=
1774:)
1771:a
1768:,
1765:s
1762:(
1759:L
1736:)
1733:n
1730:(
1727:a
1689:.
1686:)
1683:h
1677:f
1674:(
1668:)
1665:g
1659:f
1656:(
1653:=
1643:)
1639:)
1634:d
1631:n
1626:(
1622:h
1618:)
1613:d
1610:n
1605:(
1601:f
1598:(
1592:)
1589:)
1586:d
1583:(
1580:g
1577:)
1574:d
1571:(
1568:f
1565:(
1560:n
1556:|
1552:d
1544:=
1534:=
1526:f
1518:)
1514:)
1509:d
1506:n
1501:(
1497:h
1494:)
1491:d
1488:(
1485:g
1482:(
1476:)
1472:)
1467:d
1464:n
1459:(
1455:f
1452:)
1449:d
1446:(
1443:f
1440:(
1435:n
1431:|
1427:d
1419:=
1409:=
1406:)
1402:)
1397:d
1394:n
1389:(
1385:h
1382:)
1379:d
1376:(
1373:g
1370:(
1364:)
1361:n
1358:(
1355:f
1350:n
1346:|
1342:d
1334:=
1324:=
1320:)
1315:d
1312:n
1307:(
1303:h
1300:)
1297:d
1294:(
1291:g
1286:n
1282:|
1278:d
1267:)
1264:n
1261:(
1258:f
1255:=
1248:)
1245:n
1242:(
1238:)
1234:h
1228:g
1224:(
1217:f
1157:1
1154:=
1151:)
1148:n
1145:(
1142:1
1122:1
1119:=
1116:h
1113:=
1110:g
1088:f
1079:=
1076:f
1070:f
1058:f
1026:*
1010:)
1007:h
1001:f
998:(
992:)
989:g
983:f
980:(
977:=
974:)
971:h
965:g
962:(
956:f
944:f
903:f
883:f
868:q
866:(
864:f
860:p
858:(
856:f
852:n
850:(
848:f
844:q
841:p
837:n
833:n
798:f
794:c
792:(
790:f
788:)
786:b
784:(
782:f
778:c
775:b
765:(
763:f
759:a
755:a
753:(
751:f
747:n
721:)
715:,
710:+
705:Z
700:(
664:=
661:)
658:1
655:(
652:f
646:0
643:=
640:)
637:1
634:(
631:f
616:0
613:=
609:)
605:1
599:)
596:1
593:(
590:f
586:(
582:)
579:1
576:(
573:f
558:0
555:=
552:)
549:1
546:(
543:f
535:2
531:)
527:1
524:(
521:f
504:2
500:)
496:1
493:(
490:f
487:=
484:)
481:1
478:(
475:f
460:)
457:1
454:(
451:f
448:)
445:1
442:(
439:f
436:=
433:)
430:1
427:(
424:f
412:)
409:1
403:1
400:(
397:f
394:=
391:)
388:1
385:(
382:f
358:)
355:1
352:(
349:f
329:1
309:0
289:1
286:=
283:)
280:1
277:(
274:f
264:a
260:a
258:(
256:f
252:f
248:f
231:)
228:b
225:(
222:f
219:)
216:a
213:(
210:f
207:=
204:)
201:b
198:a
195:(
192:f
189:,
186:)
183:f
180:(
169:b
166:,
163:a
140:1
137:=
134:)
131:1
128:(
125:f
112:b
108:a
100:b
98:(
96:f
94:)
92:a
90:(
88:f
82:(
80:f
76:f
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.