1963:
667:
1433:
952:
of this conjecture verifies the GRH for several thousand small characters up to a certain imaginary part to obtain sufficient bounds that prove the conjecture for all integers above 10, integers below which have already been verified by calculation.
57:-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the
303:
508:
1176:
1288:
841:
773:
1010:
941:
873:
The
Ivanyos–Karpinski–Saxena deterministic algorithm for factoring polynomials over finite fields with prime constant-smooth degrees is guaranteed to run in polynomial time.
408:
690:
349:
1923:
714:
2001:
1778:
1743:
2274:
1783:
2198:
662:{\displaystyle \pi (x,a,d)={\frac {1}{\varphi (d)}}\int _{2}^{x}{\frac {1}{\ln t}}\,dt+O(x^{1/2+\varepsilon })\quad {\mbox{ as }}\ x\to \infty ,}
221:
2183:
2045:
2162:
1898:
1888:
1873:
1793:
1933:
2228:
1598:
1512:
2126:
1948:
1428:{\displaystyle {\frac {|C|}{|G|}}{\Bigl (}\operatorname {li} (x)+O{\bigl (}{\sqrt {x}}(n\log x+\log |\Delta |){\bigr )}{\Bigr )},}
1104:
1994:
351:
is primitive) defined on the whole complex plane. The generalized
Riemann hypothesis asserts that, for every Dirichlet character
856:
1943:
1938:
1913:
2167:
1736:
1459:
798:
730:
2259:
2188:
1714:
963:
945:
2092:
1763:
867:
1878:
1903:
53:, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these
2264:
1987:
1709:
1928:
1908:
1704:
693:
1534:
1267:
1243:
2279:
1967:
1729:
1918:
1808:
1773:
1474:
58:
1828:
882:
148:
859:
is guaranteed to run in polynomial time. (A polynomial-time primality test which does not require GRH, the
2157:
2078:
2053:
1843:
1571:
Ivanyos, Gabor; Karpinski, Marek; Saxena, Nitin (2009). "Schemes for deterministic polynomial factoring".
1091:
427:
136:
in 1884. Like the original
Riemann hypothesis, it has far reaching consequences about the distribution of
77:
2193:
2031:
1868:
1202:
to the whole complex plane. The resulting function encodes important information about the number field
1199:
324:
43:
108:(GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label
2269:
2121:
2036:
1464:
895:
721:
328:
1893:
1803:
1788:
151:
144:
85:
1231:
The ordinary
Riemann hypothesis follows from the extended one if one takes the number field to be
2147:
2101:
1813:
1665:
1645:
1604:
1576:
1553:
1504:
860:
89:
31:
2213:
2203:
1883:
1823:
1594:
1508:
1498:
675:
17:
1573:
Proceedings of the 2009 international symposium on
Symbolic and algebraic computation (ISAAC)
92:). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the
2131:
2083:
1818:
1798:
1752:
1635:
1586:
1543:
1494:
1054:
1044:
334:
1863:
1838:
1037:
1033:
949:
851:. This is often used in proofs, and it has many consequences, for example (assuming GRH):
1853:
1848:
717:
699:
483:
denote the number of prime numbers in this progression which are less than or equal to
423:
309:
69:
1685:
Lagarias, J.C.; Odlyzko, A.M. (1977). "Effective
Versions of the Chebotarev Theorem".
948:
also follows from the generalized
Riemann hypothesis. The yet to be verified proof of
2253:
2062:
2017:
1469:
957:
1608:
46:. Various geometrical and arithmetical objects can be described by so-called global
2238:
2233:
1833:
1029:
464:
137:
73:
1198:
The
Dedekind zeta-function satisfies a functional equation and can be extended by
1858:
133:
39:
1979:
2010:
1083:
1073:
81:
47:
35:
1503:. Graduate Texts in Mathematics. Vol. 74. Revised and with a preface by
1590:
298:{\displaystyle L(\chi ,s)=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}}
956:
Assuming the truth of the GRH, the estimate of the character sum in the
487:. If the generalized Riemann hypothesis is true, then for every coprime
1649:
1557:
1058:
727:
If GRH is true, then every proper subgroup of the multiplicative group
420:
1206:. The extended Riemann hypothesis asserts that for every number field
211:. If such a character is given, we define the corresponding Dirichlet
1185:
with real part > 1. The sum extends over all non-zero ideals
1640:
1623:
1548:
1529:
1721:
1670:
1581:
1983:
1725:
112:
to cover the extension of the
Riemann hypothesis to all global
1664:
Helfgott, Harald (2013). "Major arcs for
Goldbach's theorem".
1438:
where the constant implied in the big-O notation is absolute,
1530:"Explicit bounds for primality testing and related problems"
1171:{\displaystyle \zeta _{K}(s)=\sum _{a}{\frac {1}{(Na)^{s}}}}
888:(a generator of the multiplicative group of integers modulo
1507:(Third ed.). New York: Springer-Verlag. p. 124.
132:-functions) was probably formulated for the first time by
836:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}
768:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}
638:
1291:
1107:
966:
898:
801:
733:
702:
678:
511:
378:
is not a negative real number, then the real part of
337:
224:
1005:{\displaystyle O\left({\sqrt {q}}\log \log q\right)}
116:-functions, not just the special case of Dirichlet
2212:
2176:
2140:
2114:
2071:
2024:
1427:
1170:
1004:
935:
835:
767:
708:
684:
661:
343:
297:
143:The formal statement of the hypothesis follows. A
128:The generalized Riemann hypothesis (for Dirichlet
27:Mathematical conjecture about zeros of L-functions
1417:
1328:
1624:"Searching for primitive roots in finite fields"
1254:is a finite Galois extension with Galois group
720:. This is a considerable strengthening of the
1995:
1737:
1410:
1356:
8:
1779:Grothendieck–Hirzebruch–Riemann–Roch theorem
1242:The ERH implies an effective version of the
1228:is between 0 and 1, then it is in fact 1/2.
1082:, other than the zero ideal, we denote its
843:is generated by a set of numbers less than
42:. It is a statement about the zeros of the
2002:
1988:
1980:
1744:
1730:
1722:
157:such that there exists a positive integer
1924:Riemann–Roch theorem for smooth manifolds
1669:
1639:
1580:
1547:
1416:
1415:
1409:
1408:
1400:
1392:
1361:
1355:
1354:
1327:
1326:
1318:
1310:
1303:
1295:
1292:
1290:
1159:
1140:
1134:
1112:
1106:
975:
965:
921:
897:
827:
819:
818:
810:
806:
805:
800:
759:
751:
750:
742:
738:
737:
732:
701:
677:
637:
617:
613:
593:
575:
569:
564:
539:
510:
336:
287:
267:
261:
250:
223:
870:is guaranteed to run in polynomial time.
400:yields the ordinary Riemann hypothesis.
96:and when it is formulated for Dirichlet
1486:
877:If GRH is true, then for every prime
327:, this function can be extended to a
7:
2184:Birch and Swinnerton-Dyer conjecture
1016:being the modulus of the character.
124:Generalized Riemann hypothesis (GRH)
1889:Riemannian connection on a surface
1794:Measurable Riemann mapping theorem
1397:
1278:with Frobenius conjugacy class in
653:
262:
61:case (not the number field case).
25:
2229:Main conjecture of Iwasawa theory
1705:"Riemann hypothesis, generalized"
1020:Extended Riemann hypothesis (ERH)
783:, as well as a number coprime to
94:extended Riemann hypothesis (ERH)
2275:Unsolved problems in mathematics
1962:
1961:
1262:a union of conjugacy classes of
68:-functions can be associated to
1874:Riemann's differential equation
1784:Hirzebruch–Riemann–Roch theorem
936:{\displaystyle O((\ln p)^{6}).}
636:
100:-functions, it is known as the
88:(in which case they are called
76:(in which case they are called
2163:Ramanujan–Petersson conjecture
2153:Generalized Riemann hypothesis
2049:-functions of Hecke characters
1899:Riemann–Hilbert correspondence
1769:Generalized Riemann hypothesis
1405:
1401:
1393:
1368:
1345:
1339:
1319:
1311:
1304:
1296:
1156:
1146:
1124:
1118:
927:
918:
905:
902:
824:
802:
756:
734:
650:
633:
606:
554:
548:
533:
515:
279:
273:
240:
228:
110:generalized Riemann hypothesis
106:generalised Riemann hypothesis
102:generalized Riemann hypothesis
18:Generalized Riemann Hypothesis
1:
2122:Analytic class number formula
1934:Riemann–Siegel theta function
34:is one of the most important
2127:Riemann–von Mangoldt formula
1949:Riemann–von Mangoldt formula
1500:Multiplicative Number Theory
1710:Encyclopedia of Mathematics
1224:) = 0: if the real part of
958:Pólya–Vinogradov inequality
857:Miller–Rabin primality test
2296:
1944:Riemann–Stieltjes integral
1939:Riemann–Silberstein vector
1914:Riemann–Liouville integral
1628:Mathematics of Computation
1535:Mathematics of Computation
1450:, and Δ its discriminant.
1244:Chebotarev density theorem
946:Goldbach's weak conjecture
1957:
1879:Riemann's minimal surface
1759:
1210:and every complex number
1181:for every complex number
863:, was published in 2002.)
775:omits a number less than
355:and every complex number
149:completely multiplicative
1904:Riemann–Hilbert problems
1809:Riemann curvature tensor
1774:Grand Riemann hypothesis
1764:Cauchy–Riemann equations
1475:Grand Riemann hypothesis
1235:, with ring of integers
868:Shanks–Tonelli algorithm
694:Euler's totient function
685:{\displaystyle \varphi }
59:algebraic function field
2079:Dedekind zeta functions
1829:Riemann mapping theorem
1687:Algebraic Number Fields
1591:10.1145/1576702.1576730
78:Dedekind zeta-functions
1929:Riemann–Siegel formula
1909:Riemann–Lebesgue lemma
1844:Riemann series theorem
1622:Shoup, Victor (1992).
1429:
1172:
1092:Dedekind zeta-function
1032:(a finite-dimensional
1006:
937:
837:
769:
710:
686:
663:
428:arithmetic progression
345:
299:
266:
2199:Bloch–Kato conjecture
2194:Beilinson conjectures
2177:Algebraic conjectures
2032:Riemann zeta function
1869:Riemann zeta function
1430:
1200:analytic continuation
1173:
1007:
938:
838:
770:
711:
687:
664:
346:
344:{\displaystyle \chi }
325:analytic continuation
300:
246:
90:Dirichlet L-functions
44:Riemann zeta function
2260:Zeta and L-functions
2204:Langlands conjecture
2189:Deligne's conjecture
2141:Analytic conjectures
1919:Riemann–Roch theorem
1575:. pp. 191–198.
1465:Dirichlet L-function
1289:
1105:
964:
896:
892:) that is less than
799:
731:
722:prime number theorem
700:
676:
509:
335:
329:meromorphic function
222:
86:Dirichlet characters
2158:Lindelöf hypothesis
1894:Riemannian geometry
1804:Riemann Xi function
1789:Local zeta function
1528:Bach, Eric (1990).
1098:is then defined by
960:can be improved to
883:primitive root mod
574:
467:prime numbers. Let
409:Dirichlet's theorem
404:Consequences of GRH
152:arithmetic function
145:Dirichlet character
2265:Algebraic geometry
2148:Riemann hypothesis
2072:Algebraic examples
1814:Riemann hypothesis
1505:Hugh L. Montgomery
1460:Artin's conjecture
1425:
1168:
1139:
1053:(this ring is the
1002:
933:
861:AKS primality test
833:
795:. In other words,
765:
706:
682:
659:
642:
560:
341:
295:
32:Riemann hypothesis
2247:
2246:
2025:Analytic examples
1977:
1976:
1884:Riemannian circle
1824:Riemann invariant
1495:Davenport, Harold
1442:is the degree of
1366:
1324:
1268:unramified primes
1166:
1130:
980:
709:{\displaystyle O}
646:
641:
591:
558:
293:
16:(Redirected from
2287:
2280:Bernhard Riemann
2168:Artin conjecture
2132:Weil conjectures
2004:
1997:
1990:
1981:
1965:
1964:
1819:Riemann integral
1799:Riemann (crater)
1753:Bernhard Riemann
1746:
1739:
1732:
1723:
1718:
1691:
1690:
1682:
1676:
1675:
1673:
1660:
1654:
1653:
1643:
1634:(197): 369–380.
1619:
1613:
1612:
1584:
1568:
1562:
1561:
1551:
1542:(191): 355–380.
1525:
1519:
1518:
1491:
1434:
1432:
1431:
1426:
1421:
1420:
1414:
1413:
1404:
1396:
1367:
1362:
1360:
1359:
1332:
1331:
1325:
1323:
1322:
1314:
1308:
1307:
1299:
1293:
1266:, the number of
1177:
1175:
1174:
1169:
1167:
1165:
1164:
1163:
1141:
1138:
1117:
1116:
1055:integral closure
1045:ring of integers
1011:
1009:
1008:
1003:
1001:
997:
981:
976:
942:
940:
939:
934:
926:
925:
850:
842:
840:
839:
834:
832:
831:
822:
814:
809:
794:
782:
774:
772:
771:
766:
764:
763:
754:
746:
741:
715:
713:
712:
707:
691:
689:
688:
683:
668:
666:
665:
660:
644:
643:
639:
632:
631:
621:
592:
590:
576:
573:
568:
559:
557:
540:
501:
482:
462:
452:
442:
395:
373:
350:
348:
347:
342:
322:
304:
302:
301:
296:
294:
292:
291:
282:
268:
265:
260:
210:
198:
183:
21:
2295:
2294:
2290:
2289:
2288:
2286:
2285:
2284:
2250:
2249:
2248:
2243:
2208:
2172:
2136:
2110:
2067:
2020:
2008:
1978:
1973:
1953:
1864:Riemann surface
1839:Riemann problem
1755:
1750:
1703:
1700:
1698:Further reading
1695:
1694:
1684:
1683:
1679:
1663:
1661:
1657:
1641:10.2307/2153041
1621:
1620:
1616:
1601:
1570:
1569:
1565:
1549:10.2307/2008811
1527:
1526:
1522:
1515:
1493:
1492:
1488:
1483:
1456:
1309:
1294:
1287:
1286:
1219:
1194:
1155:
1145:
1108:
1103:
1102:
1081:
1052:
1034:field extension
1022:
974:
970:
962:
961:
950:Harald Helfgott
917:
894:
893:
881:there exists a
844:
823:
797:
796:
788:
776:
755:
729:
728:
698:
697:
674:
673:
609:
580:
544:
507:
506:
496:
468:
465:infinitely many
463:, ... contains
454:
444:
434:
424:natural numbers
411:states that if
406:
386:
360:
333:
332:
316:
283:
269:
220:
219:
200:
189:
162:
126:
70:elliptic curves
28:
23:
22:
15:
12:
11:
5:
2293:
2291:
2283:
2282:
2277:
2272:
2267:
2262:
2252:
2251:
2245:
2244:
2242:
2241:
2236:
2231:
2225:
2223:
2210:
2209:
2207:
2206:
2201:
2196:
2191:
2186:
2180:
2178:
2174:
2173:
2171:
2170:
2165:
2160:
2155:
2150:
2144:
2142:
2138:
2137:
2135:
2134:
2129:
2124:
2118:
2116:
2112:
2111:
2109:
2108:
2099:
2090:
2081:
2075:
2073:
2069:
2068:
2066:
2065:
2060:
2051:
2043:
2034:
2028:
2026:
2022:
2021:
2009:
2007:
2006:
1999:
1992:
1984:
1975:
1974:
1972:
1971:
1958:
1955:
1954:
1952:
1951:
1946:
1941:
1936:
1931:
1926:
1921:
1916:
1911:
1906:
1901:
1896:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1854:Riemann sphere
1851:
1849:Riemann solver
1846:
1841:
1836:
1831:
1826:
1821:
1816:
1811:
1806:
1801:
1796:
1791:
1786:
1781:
1776:
1771:
1766:
1760:
1757:
1756:
1751:
1749:
1748:
1741:
1734:
1726:
1720:
1719:
1699:
1696:
1693:
1692:
1677:
1655:
1614:
1599:
1563:
1520:
1513:
1485:
1484:
1482:
1479:
1478:
1477:
1472:
1467:
1462:
1455:
1452:
1436:
1435:
1424:
1419:
1412:
1407:
1403:
1399:
1395:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1365:
1358:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1330:
1321:
1317:
1313:
1306:
1302:
1298:
1274:of norm below
1215:
1190:
1179:
1178:
1162:
1158:
1154:
1151:
1148:
1144:
1137:
1133:
1129:
1126:
1123:
1120:
1115:
1111:
1077:
1048:
1021:
1018:
1000:
996:
993:
990:
987:
984:
979:
973:
969:
932:
929:
924:
920:
916:
913:
910:
907:
904:
901:
875:
874:
871:
864:
830:
826:
821:
817:
813:
808:
804:
762:
758:
753:
749:
745:
740:
736:
718:Big O notation
705:
681:
670:
669:
658:
655:
652:
649:
640: as
635:
630:
627:
624:
620:
616:
612:
608:
605:
602:
599:
596:
589:
586:
583:
579:
572:
567:
563:
556:
553:
550:
547:
543:
538:
535:
532:
529:
526:
523:
520:
517:
514:
495:and for every
405:
402:
340:
310:complex number
306:
305:
290:
286:
281:
278:
275:
272:
264:
259:
256:
253:
249:
245:
242:
239:
236:
233:
230:
227:
125:
122:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2292:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2257:
2255:
2240:
2237:
2235:
2232:
2230:
2227:
2226:
2224:
2222:
2220:
2216:
2211:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2181:
2179:
2175:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2145:
2143:
2139:
2133:
2130:
2128:
2125:
2123:
2120:
2119:
2117:
2113:
2107:
2105:
2100:
2098:
2096:
2091:
2089:
2087:
2082:
2080:
2077:
2076:
2074:
2070:
2064:
2063:Selberg class
2061:
2059:
2057:
2052:
2050:
2048:
2044:
2042:
2040:
2035:
2033:
2030:
2029:
2027:
2023:
2019:
2018:number theory
2015:
2013:
2005:
2000:
1998:
1993:
1991:
1986:
1985:
1982:
1970:
1969:
1960:
1959:
1956:
1950:
1947:
1945:
1942:
1940:
1937:
1935:
1932:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
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2254:Categories
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2058:-functions
2041:-functions
2037:Dirichlet
2014:-functions
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