1425:
and zeta functions is extensive. The idea is to extend the various arguments and techniques of arithmetic functions and zeta functions in classical analytic number theory to the context of an arbitrary arithmetic semigroup which may satisfy one or more additional axioms. Such a typical axiom is the
1697:
1600:
188:
951:
813:
463:
1404:
378:
1854:
1245:
1069:
1507:
1315:
1619:
283:
563:
501:
1455:
589:
527:
1021:
325:
1475:
1512:
101:
858:
2018:
2030:
697:
1996:
1969:
1186:
objects, i.e. objects which cannot be decomposed as a direct product of nonzero objects. Some typical examples are the following.
2057:
2062:
2014:
1749:
1768:
of the formation and the equivalence classes are generalised arithmetic progressions or generalised ideal classes. If χ is a
1748:
and allows for abstract asymptotic distribution results under constraints. In the case of number fields, for example, this is
227:, and two factorisations are considered the same if they differ only by the order of the factors indicated. The elements of
384:
1961:
967:
1351:
331:
1785:
1199:
1175:
1150:
1345:
1026:
1745:
618:
1480:
1874:
1089:
24:
835:
36:
1692:{\displaystyle \pi _{G}(x)\sim {\frac {x^{\delta }}{\delta \log x}}{\mbox{ as }}x\rightarrow \infty }
1268:
1170:
which satisfy a theorem of Krull-Schmidt type can be considered. In all these cases, the elements of
28:
1422:
1108:
1093:
685:
32:
254:
1337:
1112:
532:
2026:
1992:
1965:
1741:
1341:
470:
1440:
2036:
2002:
1975:
1777:
1258:
568:
506:
990:
294:
2040:
2006:
1979:
1769:
1460:
1407:
1097:
1072:
2051:
1255:
1193:
40:
1595:{\displaystyle N_{G}(x)=Ax^{\delta }+O(x^{\nu }){\mbox{ as }}x\rightarrow \infty .}
1248:
981:
839:
1148:. In this case, the appropriate generalisation of the prime number theorem is the
183:{\displaystyle a=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{r}^{\alpha _{r}}}
946:{\displaystyle A(x)=\exp \left({\sum _{m\geq 1}{\frac {P(x^{m})}{m}}}\right)\ .}
245:
56:
20:
1190:
27:
and applies them to a variety of different mathematical fields. The classical
963:
70:
962:
The prototypical example of an arithmetic semigroup is the multiplicative
602:"Additive number system" redirects here. For food additive numbering, see
1262:
603:
2025:. Cambridge studies in advanced mathematics. Vol. 97. p. 278.
1868:
970:
216:
1960:. Mathematical Surveys and Monographs. Vol. 86. Providence, RI:
1859:
which provides a notion of zeta function for arithmetical semigroup.
808:{\displaystyle A(x)=\sum _{n}a(n)x^{n}=\prod _{n}(1-x^{n})^{-p(n)}\ }
73:
59:
35:. The theory was invented and developed by mathematicians such as
31:
serves as a prototypical example, and the emphasis is on abstract
818:
which formally encodes the unique expression of each element of
1154:, which describes the asymptotic distribution of the ideals in
624:
If the norm is integer-valued, we associate counting functions
1329:. The indecomposable objects are the compact simply-connected
1265:
under the
Riemannian product of manifolds and norm mapping
1196:
under the usual direct product operation and norm mapping
613:
is an arithmetic semigroup in which the underlying monoid
987:= {2, 3, 5, ...}. Here, the norm of an integer is simply
1752:. An arithmetical formation is an arithmetic semigroup
1426:
following, usually called "Axiom A" in the literature:
1756:
with an equivalence relation ≡ such that the quotient
1674:
1574:
434:
355:
264:
1788:
1622:
1515:
1483:
1463:
1443:
1354:
1271:
1202:
1029:
993:
861:
700:
571:
535:
509:
473:
458:{\displaystyle |ab|=|a||b|{\mbox{ for all }}a,b\in G}
387:
334:
297:
257:
104:
1126:
forms an arithmetic semigroup with identity element
1605:For any arithmetic semigroup which satisfies Axiom
1848:
1691:
1594:
1501:
1469:
1449:
1398:
1309:
1239:
1063:
1015:
945:
852:The fundamental identity has the alternative form
807:
583:
557:
521:
495:
457:
372:
319:
277:
182:
23:which takes the ideas and techniques of classical
1399:{\displaystyle |X|=2^{\operatorname {card} (X)}.}
1137:is given by the cardinality of the quotient ring
2023:Multiplicative number theory I. Classical theory
1991:(2nd ed.). New York, NY: Dover Publishing.
621:. The norm function may be written additively.
373:{\displaystyle |p|>1{\mbox{ for all }}p\in P}
1958:Number theoretic density and logical limit laws
51:The fundamental notion involved is that of an
1849:{\displaystyle \sum _{g\in G}\chi ()|g|^{-s}}
1240:{\displaystyle |A|=\operatorname {card} (A).}
8:
1058:
1052:
1174:are isomorphism classes in an appropriate
1064:{\displaystyle N_{G}(x)=\lfloor x\rfloor }
980:= {1, 2, 3, ...}, with subset of rational
1837:
1832:
1823:
1793:
1787:
1673:
1651:
1645:
1627:
1621:
1573:
1564:
1545:
1520:
1514:
1482:
1462:
1442:
1375:
1363:
1355:
1353:
1292:
1280:
1272:
1270:
1211:
1203:
1201:
1034:
1028:
1002:
994:
992:
917:
904:
892:
887:
860:
784:
774:
755:
742:
720:
699:
570:
544:
536:
534:
508:
478:
472:
433:
428:
420:
415:
407:
399:
388:
386:
354:
343:
335:
333:
306:
298:
296:
270:
263:
258:
256:
172:
167:
162:
147:
142:
137:
125:
120:
115:
103:
1886:
1182:consists of all isomorphism classes of
92:has a unique factorisation of the form
1502:{\displaystyle 0\leq \nu <\delta }
65:satisfying the following properties:
7:
1406:The indecomposable objects are the
1247:The indecomposable objects are the
1686:
1586:
1325:denotes the manifold dimension of
14:
1871:, a property of dynamical systems
1740:provides a generalisation of the
1433:. There exist positive constants
1092:, i.e. a finite extension of the
660:counts the number of elements of
648:counts the number of elements of
1989:Abstract Analytic Number Theory
1310:{\displaystyle |M|=c^{\dim M},}
76:(finite or countably infinite)
33:asymptotic distribution results
17:Abstract analytic number theory
1833:
1824:
1820:
1817:
1811:
1808:
1683:
1639:
1633:
1583:
1570:
1557:
1532:
1526:
1388:
1382:
1364:
1356:
1281:
1273:
1261:globally symmetric Riemannian
1231:
1225:
1212:
1204:
1046:
1040:
1003:
995:
923:
910:
871:
865:
797:
791:
781:
761:
735:
729:
710:
704:
545:
537:
490:
484:
429:
421:
416:
408:
400:
389:
344:
336:
307:
299:
271:
259:
1:
1962:American Mathematical Society
1947:Knopfmacher (1990) pp.250–264
1760:/≡ is a finite abelian group
1712:) = total number of elements
1611:abstract prime number theorem
1750:Chebotarev's density theorem
822:as a product of elements of
278:{\displaystyle |{\mbox{ }}|}
1987:Knopfmacher, John (1990) .
1956:Burris, Stanley N. (2001).
2079:
1151:Landau prime ideal theorem
601:
84:, such that every element
43:in the twentieth century.
1321:> 1 is fixed, and dim
1133:and the norm of an ideal
565:is finite, for each real
558:{\displaystyle |a|\leq x}
205:are distinct elements of
1938:Knopfmacher (1990) p.154
1764:. This quotient is the
1609:, we have the following
496:{\displaystyle N_{G}(x)}
2058:Algebraic number theory
1929:Knopfmacher (1990) p.75
1746:algebraic number theory
1450:{\displaystyle \delta }
1168:arithmetical categories
684:) be the corresponding
598:Additive number systems
2063:Analytic number theory
1875:Beurling zeta function
1850:
1738:arithmetical formation
1732:Arithmetical formation
1693:
1596:
1503:
1471:
1451:
1417:Methods and techniques
1400:
1311:
1241:
1090:algebraic number field
1065:
1017:
947:
809:
611:additive number system
585:
584:{\displaystyle x>0}
559:
523:
522:{\displaystyle a\in G}
497:
459:
374:
321:
279:
184:
25:analytic number theory
1851:
1776:then we can define a
1694:
1597:
1504:
1472:
1452:
1401:
1312:
1251:of prime power order.
1242:
1066:
1018:
1016:{\displaystyle |n|=n}
948:
836:radius of convergence
828:radius of convergence
810:
586:
560:
524:
498:
460:
375:
322:
320:{\displaystyle |1|=1}
280:
185:
47:Arithmetic semigroups
1786:
1620:
1513:
1481:
1470:{\displaystyle \nu }
1461:
1441:
1423:arithmetic functions
1352:
1336:The category of all
1269:
1254:The category of all
1200:
1189:The category of all
1027:
991:
859:
698:
690:fundamental identity
569:
533:
507:
471:
385:
332:
295:
255:
102:
53:arithmetic semigroup
29:prime number theorem
2015:Montgomery, Hugh L.
686:formal power series
436: for all
357: for all
179:
154:
132:
2019:Vaughan, Robert C.
1920:Burris (2001) p.34
1911:Burris (2001) p.31
1902:Burris (2001) p.26
1893:Burris (2001) p.20
1846:
1804:
1689:
1678:
1592:
1578:
1499:
1467:
1457:, and a constant
1447:
1396:
1342:topological spaces
1307:
1237:
1061:
1013:
943:
903:
805:
760:
725:
581:
555:
519:
493:
455:
438:
370:
359:
317:
275:
268:
180:
158:
133:
111:
2032:978-0-521-84903-6
1789:
1742:ideal class group
1677:
1671:
1577:
1348:and norm mapping
1333:symmetric spaces.
939:
930:
888:
804:
751:
716:
467:The total number
437:
358:
267:
2070:
2044:
2010:
1983:
1948:
1945:
1939:
1936:
1930:
1927:
1921:
1918:
1912:
1909:
1903:
1900:
1894:
1891:
1855:
1853:
1852:
1847:
1845:
1844:
1836:
1827:
1803:
1778:Dirichlet series
1698:
1696:
1695:
1690:
1679:
1675:
1672:
1670:
1656:
1655:
1646:
1632:
1631:
1601:
1599:
1598:
1593:
1579:
1575:
1569:
1568:
1550:
1549:
1525:
1524:
1508:
1506:
1505:
1500:
1476:
1474:
1473:
1468:
1456:
1454:
1453:
1448:
1408:connected spaces
1405:
1403:
1402:
1397:
1392:
1391:
1367:
1359:
1338:pseudometrisable
1316:
1314:
1313:
1308:
1303:
1302:
1284:
1276:
1259:simply-connected
1246:
1244:
1243:
1238:
1215:
1207:
1098:rational numbers
1073:greatest integer
1070:
1068:
1067:
1062:
1039:
1038:
1022:
1020:
1019:
1014:
1006:
998:
952:
950:
949:
944:
937:
936:
932:
931:
926:
922:
921:
905:
902:
814:
812:
811:
806:
802:
801:
800:
779:
778:
759:
747:
746:
724:
590:
588:
587:
582:
564:
562:
561:
556:
548:
540:
528:
526:
525:
520:
502:
500:
499:
494:
483:
482:
464:
462:
461:
456:
439:
435:
432:
424:
419:
411:
403:
392:
379:
377:
376:
371:
360:
356:
347:
339:
326:
324:
323:
318:
310:
302:
284:
282:
281:
276:
274:
269:
265:
262:
189:
187:
186:
181:
178:
177:
176:
166:
153:
152:
151:
141:
131:
130:
129:
119:
37:John Knopfmacher
2078:
2077:
2073:
2072:
2071:
2069:
2068:
2067:
2048:
2047:
2033:
2013:
1999:
1986:
1972:
1955:
1952:
1951:
1946:
1942:
1937:
1933:
1928:
1924:
1919:
1915:
1910:
1906:
1901:
1897:
1892:
1888:
1883:
1865:
1831:
1784:
1783:
1734:
1707:
1657:
1647:
1623:
1618:
1617:
1560:
1541:
1516:
1511:
1510:
1479:
1478:
1459:
1458:
1439:
1438:
1419:
1371:
1350:
1349:
1346:topological sum
1288:
1267:
1266:
1198:
1197:
1159:
1142:
1131:
1120:
1107:of all nonzero
1103:, then the set
1030:
1025:
1024:
989:
988:
959:
913:
906:
883:
857:
856:
780:
770:
738:
696:
695:
688:. We have the
607:
600:
567:
566:
531:
530:
505:
504:
474:
469:
468:
383:
382:
330:
329:
293:
292:
253:
252:
244:There exists a
231:are called the
214:
204:
168:
143:
121:
100:
99:
69:There exists a
49:
19:is a branch of
12:
11:
5:
2076:
2074:
2066:
2065:
2060:
2050:
2049:
2046:
2045:
2031:
2011:
1997:
1984:
1970:
1950:
1949:
1940:
1931:
1922:
1913:
1904:
1895:
1885:
1884:
1882:
1879:
1878:
1877:
1872:
1864:
1861:
1857:
1856:
1843:
1840:
1835:
1830:
1826:
1822:
1819:
1816:
1813:
1810:
1807:
1802:
1799:
1796:
1792:
1736:The notion of
1733:
1730:
1703:
1700:
1699:
1688:
1685:
1682:
1676: as
1669:
1666:
1663:
1660:
1654:
1650:
1644:
1641:
1638:
1635:
1630:
1626:
1603:
1602:
1591:
1588:
1585:
1582:
1576: as
1572:
1567:
1563:
1559:
1556:
1553:
1548:
1544:
1540:
1537:
1534:
1531:
1528:
1523:
1519:
1498:
1495:
1492:
1489:
1486:
1466:
1446:
1418:
1415:
1414:
1413:
1412:
1411:
1395:
1390:
1387:
1384:
1381:
1378:
1374:
1370:
1366:
1362:
1358:
1334:
1306:
1301:
1298:
1295:
1291:
1287:
1283:
1279:
1275:
1252:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1214:
1210:
1206:
1194:abelian groups
1184:indecomposable
1163:
1162:
1157:
1140:
1129:
1118:
1081:
1080:
1075:not exceeding
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1037:
1033:
1012:
1009:
1005:
1001:
997:
958:
955:
954:
953:
942:
935:
929:
925:
920:
916:
912:
909:
901:
898:
895:
891:
886:
882:
879:
876:
873:
870:
867:
864:
816:
815:
799:
796:
793:
790:
787:
783:
777:
773:
769:
766:
763:
758:
754:
750:
745:
741:
737:
734:
731:
728:
723:
719:
715:
712:
709:
706:
703:
599:
596:
595:
594:
593:
592:
580:
577:
574:
554:
551:
547:
543:
539:
518:
515:
512:
492:
489:
486:
481:
477:
465:
454:
451:
448:
445:
442:
431:
427:
423:
418:
414:
410:
406:
402:
398:
395:
391:
380:
369:
366:
363:
353:
350:
346:
342:
338:
327:
316:
313:
309:
305:
301:
273:
261:
241:
240:
223:may depend on
210:
200:
193:
192:
191:
190:
175:
171:
165:
161:
157:
150:
146:
140:
136:
128:
124:
118:
114:
110:
107:
94:
93:
48:
45:
13:
10:
9:
6:
4:
3:
2:
2075:
2064:
2061:
2059:
2056:
2055:
2053:
2042:
2038:
2034:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1998:0-486-66344-2
1994:
1990:
1985:
1981:
1977:
1973:
1971:0-8218-2666-2
1967:
1963:
1959:
1954:
1953:
1944:
1941:
1935:
1932:
1926:
1923:
1917:
1914:
1908:
1905:
1899:
1896:
1890:
1887:
1880:
1876:
1873:
1870:
1867:
1866:
1862:
1860:
1841:
1838:
1828:
1814:
1805:
1800:
1797:
1794:
1790:
1782:
1781:
1780:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1731:
1729:
1727:
1723:
1719:
1715:
1711:
1706:
1680:
1667:
1664:
1661:
1658:
1652:
1648:
1642:
1636:
1628:
1624:
1616:
1615:
1614:
1612:
1608:
1589:
1580:
1565:
1561:
1554:
1551:
1546:
1542:
1538:
1535:
1529:
1521:
1517:
1496:
1493:
1490:
1487:
1484:
1464:
1444:
1436:
1432:
1429:
1428:
1427:
1424:
1416:
1409:
1393:
1385:
1379:
1376:
1372:
1368:
1360:
1347:
1343:
1339:
1335:
1332:
1328:
1324:
1320:
1304:
1299:
1296:
1293:
1289:
1285:
1277:
1264:
1260:
1257:
1253:
1250:
1249:cyclic groups
1234:
1228:
1222:
1219:
1216:
1208:
1195:
1192:
1188:
1187:
1185:
1181:
1177:
1173:
1169:
1165:
1164:
1160:
1153:
1152:
1147:
1143:
1136:
1132:
1125:
1121:
1114:
1110:
1106:
1102:
1099:
1095:
1091:
1087:
1083:
1082:
1078:
1074:
1055:
1049:
1043:
1035:
1031:
1010:
1007:
999:
986:
983:
979:
975:
972:
969:
965:
961:
960:
956:
940:
933:
927:
918:
914:
907:
899:
896:
893:
889:
884:
880:
877:
874:
868:
862:
855:
854:
853:
850:
848:
844:
841:
837:
833:
829:
825:
821:
794:
788:
785:
775:
771:
767:
764:
756:
752:
748:
743:
739:
732:
726:
721:
717:
713:
707:
701:
694:
693:
692:
691:
687:
683:
679:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
635:
631:
627:
622:
620:
616:
612:
605:
597:
578:
575:
572:
552:
549:
541:
516:
513:
510:
487:
479:
475:
466:
452:
449:
446:
443:
440:
425:
412:
404:
396:
393:
381:
367:
364:
361:
351:
348:
340:
328:
314:
311:
303:
291:
290:
288:
251:
247:
243:
242:
238:
234:
230:
226:
222:
218:
215:are positive
213:
208:
203:
199:
195:
194:
173:
169:
163:
159:
155:
148:
144:
138:
134:
126:
122:
116:
112:
108:
105:
98:
97:
96:
95:
91:
87:
83:
79:
75:
72:
68:
67:
66:
64:
61:
58:
55:, which is a
54:
46:
44:
42:
41:Arne Beurling
38:
34:
30:
26:
22:
18:
2022:
1988:
1957:
1943:
1934:
1925:
1916:
1907:
1898:
1889:
1858:
1773:
1765:
1761:
1757:
1753:
1737:
1735:
1725:
1721:
1717:
1713:
1709:
1704:
1701:
1610:
1606:
1604:
1509:, such that
1434:
1430:
1420:
1330:
1326:
1322:
1318:
1183:
1179:
1171:
1167:
1155:
1149:
1145:
1138:
1134:
1127:
1123:
1116:
1115:of integers
1104:
1100:
1085:
1076:
984:
977:
973:
851:
846:
842:
840:power series
831:
827:
823:
819:
817:
689:
681:
677:
673:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
625:
623:
619:free abelian
614:
610:
608:
503:of elements
286:
250:norm mapping
249:
236:
232:
228:
224:
220:
211:
206:
201:
197:
89:
85:
81:
77:
62:
52:
50:
16:
15:
1766:class group
1421:The use of
1331:irreducible
57:commutative
21:mathematics
2052:Categories
2041:1142.11001
2007:0743.11002
1980:0995.11001
1881:References
1344:under the
1023:, so that
668:. We let
289:such that
196:where the
1839:−
1806:χ
1798:∈
1791:∑
1770:character
1720:of norm |
1687:∞
1684:→
1665:
1659:δ
1653:δ
1643:∼
1625:π
1587:∞
1584:→
1566:ν
1547:δ
1497:δ
1491:ν
1488:≤
1465:ν
1445:δ
1380:
1297:
1263:manifolds
1223:
1059:⌋
1053:⌊
964:semigroup
897:≥
890:∑
881:
786:−
768:−
753:∏
718:∑
550:≤
514:∈
450:∈
365:∈
170:α
156:⋯
145:α
123:α
71:countable
2021:(2007).
1863:See also
1176:category
1166:Various
971:integers
968:positive
957:Examples
664:of norm
652:of norm
604:E number
529:of norm
248:-valued
217:integers
1869:Axiom A
1702:where π
1431:Axiom A
1340:finite
1256:compact
1111:in the
838:of the
834:is the
826:. The
640:) with
209:, the α
88:≠ 1 in
2039:
2029:
2005:
1995:
1978:
1968:
1317:where
1191:finite
1178:, and
1109:ideals
1088:is an
1071:, the
982:primes
938:
803:
676:) and
656:, and
644:where
632:) and
266:
233:primes
74:subset
60:monoid
1477:with
1094:field
2027:ISBN
1993:ISBN
1966:ISBN
1724:| ≤
1494:<
1437:and
1377:card
1220:card
1113:ring
576:>
349:>
246:real
39:and
2037:Zbl
2003:Zbl
1976:Zbl
1772:of
1744:in
1716:in
1662:log
1294:dim
1122:of
1096:of
1084:If
966:of
878:exp
849:).
830:of
617:is
609:An
285:on
235:of
80:of
2054::
2035:.
2017:;
2001:.
1974:.
1964:.
1728:.
1613::
976:=
219:,
2043:.
2009:.
1982:.
1842:s
1834:|
1829:g
1825:|
1821:)
1818:]
1815:g
1812:[
1809:(
1801:G
1795:g
1774:A
1762:A
1758:G
1754:G
1726:x
1722:p
1718:P
1714:p
1710:x
1708:(
1705:G
1681:x
1668:x
1649:x
1640:)
1637:x
1634:(
1629:G
1607:A
1590:.
1581:x
1571:)
1562:x
1558:(
1555:O
1552:+
1543:x
1539:A
1536:=
1533:)
1530:x
1527:(
1522:G
1518:N
1485:0
1435:A
1410:.
1394:.
1389:)
1386:X
1383:(
1373:2
1369:=
1365:|
1361:X
1357:|
1327:M
1323:M
1319:c
1305:,
1300:M
1290:c
1286:=
1282:|
1278:M
1274:|
1235:.
1232:)
1229:A
1226:(
1217:=
1213:|
1209:A
1205:|
1180:P
1172:G
1161:.
1158:K
1156:O
1146:I
1144:/
1141:K
1139:O
1135:I
1130:K
1128:O
1124:K
1119:K
1117:O
1105:G
1101:Q
1086:K
1079:.
1077:x
1056:x
1050:=
1047:)
1044:x
1041:(
1036:G
1032:N
1011:n
1008:=
1004:|
1000:n
996:|
985:P
978:Z
974:G
941:.
934:)
928:m
924:)
919:m
915:x
911:(
908:P
900:1
894:m
885:(
875:=
872:)
869:x
866:(
863:A
847:x
845:(
843:A
832:G
824:P
820:G
798:)
795:n
792:(
789:p
782:)
776:n
772:x
765:1
762:(
757:n
749:=
744:n
740:x
736:)
733:n
730:(
727:a
722:n
714:=
711:)
708:x
705:(
702:A
682:x
680:(
678:P
674:x
672:(
670:A
666:n
662:G
658:a
654:n
650:P
646:p
642:G
638:n
636:(
634:p
630:n
628:(
626:a
615:G
606:.
591:.
579:0
573:x
553:x
546:|
542:a
538:|
517:G
511:a
491:)
488:x
485:(
480:G
476:N
453:G
447:b
444:,
441:a
430:|
426:b
422:|
417:|
413:a
409:|
405:=
401:|
397:b
394:a
390:|
368:P
362:p
352:1
345:|
341:p
337:|
315:1
312:=
308:|
304:1
300:|
287:G
272:|
260:|
239:.
237:G
229:P
225:a
221:r
212:i
207:P
202:i
198:p
174:r
164:r
160:p
149:2
139:2
135:p
127:1
117:1
113:p
109:=
106:a
90:G
86:a
82:G
78:P
63:G
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