2441:
1035:
suspected Apéry was on to something and set out to confirm his proof. Two months later they finished verification of Apéry's proof, and on August 18 Cohen delivered a lecture giving full details of the proof. After the lecture Apéry himself took to the podium to explain the source of some of his
1022:
were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π. Due to the wholly unexpected nature of the proof and Apéry's blasé and very sketchy approach to the subject, many of the mathematicians in the audience
3369:
spanned by values of the zeta function at odd integers. Hopes that
Zudilin could cut his list further to just one number did not materialise, but work on this problem is still an active area of research. Higher zeta constants have application to physics: they describe correlation functions in
185:
1598:
519:
2649:
1328:
3077:
2206:
1824:
2908:
1950:
1150:
45:
706:
959:
Because of this, no proof could be found to show that the zeta constants with odd arguments were irrational, even though they were (and still are) all believed to be transcendental. However, in June 1978,
1404:
386:
921:
2751:
is rational by constructing sequences that tend to zero but are bounded below by some positive constant. They are somewhat less transparent than the earlier proofs, since they rely upon
1994:
2542:
2194:
1197:
2709:
2534:
3243:
2682:
2436:{\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {-\log(xy)}{1-xy}}{\tilde {P_{n}}}(x){\tilde {P_{n}}}(y)dxdy={\frac {A_{n}+B_{n}\zeta (3)}{\operatorname {lcm} \left^{3}}}}
848:
3359:
2946:
799:
582:
3330:
3301:
3272:
3136:
2800:
2749:
2507:
2023:
1396:
1189:
1020:
991:
3198:
3163:
3107:
2938:
767:
1688:
1367:
954:
359:
292:
2130:
2103:
2050:
1655:
1628:
736:
1066:
333:
2766:
2076:
550:
216:
266:
1696:
2808:
2472:
2462:
1835:
1086:
180:{\displaystyle \zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.2020569\ldots }
590:
3392:
3390:
Kohnen, Winfried (1989). "Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms".
3826:
1593:{\displaystyle c_{n,k}=\sum _{m=1}^{n}{\frac {1}{m^{3}}}+\sum _{m=1}^{k}{\frac {(-1)^{m-1}}{2m^{3}{\binom {n}{m}}{\binom {n+m}{m}}}}.}
3170:
3138:
is irrational. Unfortunately, extensive computer searching has failed to find such a constant, and in fact it is now known that if
514:{\displaystyle {\frac {1}{1^{2n}}}+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots ={\frac {p}{q}}\pi ^{2n}}
3435:
1045:
3777:
3821:
2719:
1024:
860:
2149:
3361:
must be irrational. Their work uses linear forms in values of the zeta function and estimates upon them to bound the
3632:
Rivoal, T. (2000). "La fonction zeta de
Riemann prend une infinité de valeurs irrationnelles aux entiers impairs".
3371:
3362:
2644:{\displaystyle 0<{\frac {1}{b}}\leq \left|A_{n}+B_{n}\zeta (3)\right|\leq 4\left({\frac {4}{5}}\right)^{n}}
2197:
2132:
by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.
1958:
2655:
1032:
2154:
1323:{\displaystyle \zeta (3)={\frac {5}{2}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{3}{\binom {2n}{n}}}}.}
964:
gave a talk titled "Sur l'irrationalité de ζ(3)." During the course of the talk he outlined proofs that
32:
3720:
H. E. Boos; V. E. Korepin; Y. Nishiyama; M. Shiroishi (2002). "Quantum
Correlations and Number Theory".
2752:
927:
238:
3739:
3694:
3651:
303:
242:
3200:, so extending Apéry's proof to work on the higher odd zeta constants does not seem likely to work.
3109:
were found then the methods used to prove Apéry's theorem would be expected to work on a proof that
2687:
2512:
3493:
Apéry, R. (1981), "Interpolation de fractions continues et irrationalité de certaines constantes",
3072:{\displaystyle \zeta (5)=\xi _{5}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{5}{\binom {2n}{n}}}}.}
3210:
2661:
815:
3800:
3755:
3729:
3667:
3641:
3601:
3536:
3452:
3408:
3335:
772:
555:
3306:
3277:
3248:
3112:
2776:
2725:
2483:
1999:
1372:
1165:
996:
967:
3176:
3141:
3085:
2916:
2718:
is more reminiscent of Apéry's original proof, and also has similarities to a fourth proof by
745:
739:
36:
1660:
1339:
933:
3792:
3747:
3702:
3659:
3593:
3571:
3517:
3444:
3400:
3166:
338:
295:
271:
3427:
2108:
2081:
2028:
1633:
1606:
714:
1051:
1028:
309:
299:
2055:
527:
193:
3743:
3698:
3655:
248:
961:
809:
373:
231:
3751:
3663:
3815:
3759:
3671:
3605:
3456:
3412:
3204:
2715:
2141:
298:
to be irrational, while it remains open whether the function's values are in general
28:
3706:
1819:{\displaystyle a_{n}=\sum _{k=0}^{n}c_{n,k}{\binom {n}{k}}^{2}{\binom {n+k}{k}}^{2}}
3366:
2903:{\displaystyle \zeta (2)=3\sum _{n=1}^{\infty }{\frac {1}{n^{2}{\binom {2n}{n}}}}.}
1044:
Apéry's original proof was based on the well known irrationality criterion from
20:
3685:
W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational".
1945:{\displaystyle b_{n}=\sum _{k=0}^{n}{\binom {n}{k}}^{2}{\binom {n+k}{k}}^{2}.}
1145:{\displaystyle \left|\xi -{\frac {p}{q}}\right|<{\frac {c}{q^{1+\delta }}}}
362:
3521:
3471:
2767:
Particular values of the
Riemann zeta function § Odd positive integers
2722:. These later proofs again derive a contradiction from the assumption that
3618:
D. H. Bailey, J. Borwein, N. Calkin, R. Girgensohn, R. Luke, and V. Moll,
2913:
Due to the success of Apéry's method a search was undertaken for a number
3734:
2145:
1334:
701:{\displaystyle \zeta (2n)=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}
3804:
3597:
3448:
3404:
1069:
227:
3646:
3541:
3796:
3576:
3508:
F. Beukers (1979). "A note on the irrationality of ζ(2) and ζ(3)".
2140:
Within a year of Apéry's result an alternative proof was found by
3778:"Similarities in Irrationality Proofs for π, ln2, ζ(2), and ζ(3)"
3207:
and Tanguy Rivoal has shown that infinitely many of the numbers
808:
No such representation in terms of π is known for the so-called
16:
Sum of the inverses of the positive integers cubed is irrational
3557:
3535:
Zudilin, W. (2002). "An
Elementary Proof of Apéry's Theorem".
3245:
must be irrational, and even that at least one of the numbers
1162:
The starting point for Apéry was the series representation of
854:. It has been conjectured that the ratios of these quantities
2196:. Using a representation that would later be generalized to
2476:
2466:
552:. Specifically, writing the infinite series on the left as
2078:. Nevertheless, Apéry showed that even after multiplying
2773:
Apéry and
Beukers could simplify their proofs to work on
2025:
fast enough to apply the criterion, but unfortunately
3338:
3309:
3280:
3251:
3213:
3179:
3144:
3115:
3088:
2949:
2919:
2811:
2779:
2728:
2690:
2664:
2545:
2515:
2486:
2480:). Using partial integration and the assumption that
2209:
2157:
2111:
2084:
2058:
2031:
2002:
1961:
1838:
1699:
1663:
1636:
1609:
1407:
1375:
1342:
1200:
1168:
1089:
1054:
999:
970:
936:
863:
818:
775:
748:
717:
593:
558:
530:
389:
341:
312:
274:
251:
196:
48:
916:{\displaystyle {\frac {\zeta (2n+1)}{\pi ^{2n+1}}},}
3584:Yu. V. Nesterenko (1996). "A Few Remarks on ζ(3)".
3169:of degree at most 25, then the coefficients in its
3634:Comptes Rendus de l'Académie des Sciences, Série I
3495:Bulletin de la section des sciences du C.T.H.S III
3353:
3324:
3295:
3266:
3237:
3192:
3157:
3130:
3101:
3071:
2932:
2902:
2794:
2743:
2703:
2676:
2643:
2528:
2501:
2435:
2188:
2124:
2097:
2070:
2044:
2017:
1988:
1944:
1818:
1682:
1649:
1622:
1592:
1390:
1361:
1322:
1183:
1144:
1060:
1014:
985:
948:
915:
842:
793:
761:
730:
700:
576:
544:
513:
353:
327:
286:
260:
210:
179:
2658:since the right-most expression tends to zero as
1927:
1906:
1890:
1877:
1804:
1783:
1767:
1754:
1398:about as fast as the above series, specifically
3057:
3039:
2888:
2870:
1578:
1557:
1548:
1535:
1308:
1290:
8:
2802:as well thanks to the series representation
2536:, Beukers eventually derived the inequality
3510:Bulletin of the London Mathematical Society
1068:is irrational if there are infinitely many
3733:
3645:
3575:
3540:
3337:
3308:
3279:
3250:
3212:
3184:
3178:
3149:
3143:
3114:
3093:
3087:
3056:
3038:
3036:
3030:
3012:
2996:
2990:
2979:
2969:
2948:
2924:
2918:
2887:
2869:
2867:
2861:
2851:
2845:
2834:
2810:
2778:
2727:
2691:
2689:
2663:
2635:
2621:
2587:
2574:
2552:
2544:
2516:
2514:
2485:
2424:
2372:
2359:
2352:
2316:
2310:
2309:
2288:
2282:
2281:
2240:
2234:
2229:
2219:
2214:
2208:
2165:
2159:
2158:
2156:
2116:
2110:
2089:
2083:
2057:
2036:
2030:
2001:
1978:
1968:
1962:
1960:
1933:
1926:
1905:
1903:
1896:
1889:
1876:
1874:
1867:
1856:
1843:
1837:
1810:
1803:
1782:
1780:
1773:
1766:
1753:
1751:
1738:
1728:
1717:
1704:
1698:
1668:
1662:
1641:
1635:
1614:
1608:
1577:
1556:
1554:
1547:
1534:
1532:
1526:
1505:
1489:
1483:
1472:
1457:
1448:
1442:
1431:
1412:
1406:
1374:
1347:
1341:
1307:
1289:
1287:
1281:
1263:
1247:
1241:
1230:
1216:
1199:
1167:
1128:
1119:
1101:
1088:
1053:
998:
969:
935:
893:
864:
862:
817:
774:
753:
747:
722:
716:
666:
644:
637:
625:
592:
557:
534:
529:
502:
488:
468:
459:
445:
436:
422:
413:
399:
390:
388:
340:
311:
273:
250:
200:
195:
154:
145:
134:
125:
114:
105:
94:
85:
79:
68:
47:
801:is irrational for all positive integers
769:is always irrational, this showed that
3382:
1333:Roughly speaking, Apéry then defined a
1023:dismissed the proof as flawed. However
1989:{\displaystyle {\frac {a_{n}}{b_{n}}}}
7:
2684:, and so must eventually fall below
2189:{\displaystyle {\tilde {P_{n}}}(x)}
2144:, who replaced Apéry's series with
1603:He then defined two more sequences
3620:Experimental Mathematics in Action
3393:Proc. Indian Acad. Sci. Math. Sci.
3043:
2991:
2874:
2846:
2671:
1910:
1881:
1787:
1758:
1561:
1539:
1294:
1242:
80:
14:
1657:that, roughly, have the quotient
190:cannot be written as a fraction
3707:10.1070/RM2001v056n04ABEH000427
3472:"Irrationalité de ζ(2) et ζ(3)"
3436:The Mathematical Intelligencer
3428:"A proof that Euler missed..."
3348:
3342:
3319:
3313:
3290:
3284:
3261:
3255:
3232:
3217:
3125:
3119:
3009:
2999:
2959:
2953:
2821:
2815:
2789:
2783:
2738:
2732:
2704:{\displaystyle {\frac {1}{b}}}
2668:
2602:
2596:
2529:{\displaystyle {\frac {a}{b}}}
2496:
2490:
2387:
2381:
2334:
2328:
2322:
2306:
2300:
2294:
2261:
2252:
2183:
2177:
2171:
2012:
2006:
1502:
1492:
1385:
1379:
1260:
1250:
1210:
1204:
1178:
1172:
1046:Peter Gustav Lejeune Dirichlet
1009:
1003:
980:
974:
885:
870:
837:
822:
812:for odd arguments, the values
788:
779:
689:
680:
663:
653:
622:
612:
606:
597:
571:
562:
58:
52:
1:
3664:10.1016/S0764-4442(00)01624-4
1048:, which states that a number
230:. The theorem is named after
3238:{\displaystyle \zeta (2n+1)}
2677:{\displaystyle n\to \infty }
2150:shifted Legendre polynomials
843:{\displaystyle \zeta (2n+1)}
3752:10.1088/0305-4470/35/20/305
3426:A. van der Poorten (1979).
3173:must be enormous, at least
380:is a positive integer then
294:) can be shown in terms of
3843:
3776:Huylebrouck, Dirk (2001).
3559:Некоторые замечания о ζ(3)
3354:{\displaystyle \zeta (11)}
2509:was rational and equal to
794:{\displaystyle \zeta (2n)}
742:. Once it was proved that
577:{\displaystyle \zeta (2n)}
237:The special values of the
3827:Theorems in number theory
3556:Ю. В. Нестеренко (1996).
3325:{\displaystyle \zeta (9)}
3296:{\displaystyle \zeta (7)}
3267:{\displaystyle \zeta (5)}
3131:{\displaystyle \zeta (5)}
2795:{\displaystyle \zeta (2)}
2744:{\displaystyle \zeta (3)}
2502:{\displaystyle \zeta (3)}
2018:{\displaystyle \zeta (3)}
1391:{\displaystyle \zeta (3)}
1184:{\displaystyle \zeta (3)}
1015:{\displaystyle \zeta (2)}
986:{\displaystyle \zeta (3)}
524:for some rational number
3558:
3193:{\displaystyle 10^{383}}
3158:{\displaystyle \xi _{5}}
3102:{\displaystyle \xi _{5}}
2933:{\displaystyle \xi _{5}}
2052:is not an integer after
1690:. These sequences were
762:{\displaystyle \pi ^{n}}
3165:exists and if it is an
2940:with the property that
2714:A more recent proof by
1683:{\displaystyle c_{n,k}}
1362:{\displaystyle c_{n,k}}
949:{\displaystyle n\geq 1}
3355:
3326:
3297:
3268:
3239:
3194:
3159:
3132:
3103:
3073:
2995:
2934:
2904:
2850:
2796:
2745:
2705:
2678:
2645:
2530:
2503:
2437:
2200:, Beukers showed that
2190:
2126:
2099:
2072:
2046:
2019:
1990:
1946:
1872:
1820:
1733:
1684:
1651:
1624:
1594:
1488:
1447:
1392:
1363:
1324:
1246:
1185:
1146:
1062:
1033:Alfred van der Poorten
1016:
987:
950:
917:
850:for positive integers
844:
795:
763:
732:
702:
578:
546:
515:
355:
354:{\displaystyle n>1}
329:
288:
287:{\displaystyle n>0}
262:
212:
181:
84:
39:. That is, the number
3582:English translation:
3522:10.1112/blms/11.3.268
3356:
3327:
3298:
3269:
3240:
3195:
3160:
3133:
3104:
3074:
2975:
2935:
2905:
2830:
2797:
2759:Higher zeta constants
2753:hypergeometric series
2746:
2706:
2679:
2646:
2531:
2504:
2438:
2198:Hadjicostas's formula
2191:
2127:
2125:{\displaystyle b_{n}}
2100:
2098:{\displaystyle a_{n}}
2073:
2047:
2045:{\displaystyle a_{n}}
2020:
1991:
1947:
1852:
1821:
1713:
1685:
1652:
1650:{\displaystyle b_{n}}
1625:
1623:{\displaystyle a_{n}}
1595:
1468:
1427:
1393:
1364:
1325:
1226:
1186:
1147:
1063:
1039:
1017:
988:
951:
918:
845:
796:
764:
733:
731:{\displaystyle B_{n}}
703:
579:
547:
516:
356:
330:
289:
263:
239:Riemann zeta function
213:
182:
64:
3822:Zeta and L-functions
3722:Journal of Physics A
3336:
3307:
3278:
3249:
3211:
3177:
3142:
3113:
3086:
2947:
2917:
2809:
2777:
2726:
2688:
2662:
2543:
2513:
2484:
2207:
2155:
2109:
2082:
2056:
2029:
2000:
1959:
1836:
1697:
1661:
1634:
1607:
1405:
1373:
1340:
1198:
1166:
1087:
1061:{\displaystyle \xi }
1052:
997:
968:
934:
861:
816:
773:
746:
715:
591:
556:
528:
387:
339:
328:{\displaystyle 2n+1}
310:
272:
249:
194:
46:
3785:Amer. Math. Monthly
3744:2002JPhA...35.4443B
3699:2001RuMaS..56..774Z
3656:2000CRASM.331..267R
3372:quantum spin chains
2239:
2224:
2071:{\displaystyle n=2}
1369:which converges to
545:{\displaystyle p/q}
365:to be irrational).
361:) (though they are
211:{\displaystyle p/q}
3598:10.1007/BF02307212
3470:Apéry, R. (1979).
3449:10.1007/BF03028234
3405:10.1007/BF02864395
3351:
3322:
3293:
3264:
3235:
3190:
3171:minimal polynomial
3155:
3128:
3099:
3069:
2930:
2900:
2792:
2741:
2701:
2674:
2641:
2526:
2499:
2446:for some integers
2433:
2225:
2210:
2186:
2122:
2095:
2068:
2042:
2015:
1986:
1942:
1816:
1680:
1647:
1620:
1590:
1388:
1359:
1320:
1181:
1142:
1058:
1012:
983:
946:
930:for every integer
913:
840:
791:
759:
728:
698:
574:
542:
511:
351:
325:
284:
261:{\displaystyle 2n}
258:
208:
177:
3728:(20): 4443–4452.
3064:
3055:
2895:
2886:
2699:
2629:
2560:
2524:
2431:
2325:
2297:
2279:
2174:
1984:
1925:
1888:
1802:
1765:
1585:
1576:
1546:
1463:
1315:
1306:
1224:
1140:
1109:
908:
740:Bernoulli numbers
738:are the rational
696:
496:
477:
454:
431:
408:
296:Bernoulli numbers
160:
140:
120:
100:
3834:
3808:
3782:
3764:
3763:
3737:
3735:cond-mat/0202346
3717:
3711:
3710:
3687:Russ. Math. Surv
3682:
3676:
3675:
3649:
3629:
3623:
3616:
3610:
3609:
3581:
3579:
3553:
3547:
3546:
3544:
3532:
3526:
3525:
3505:
3499:
3498:
3497:, pp. 37–53
3490:
3484:
3483:
3467:
3461:
3460:
3432:
3423:
3417:
3416:
3387:
3360:
3358:
3357:
3352:
3331:
3329:
3328:
3323:
3302:
3300:
3299:
3294:
3273:
3271:
3270:
3265:
3244:
3242:
3241:
3236:
3199:
3197:
3196:
3191:
3189:
3188:
3167:algebraic number
3164:
3162:
3161:
3156:
3154:
3153:
3137:
3135:
3134:
3129:
3108:
3106:
3105:
3100:
3098:
3097:
3078:
3076:
3075:
3070:
3065:
3063:
3062:
3061:
3060:
3051:
3042:
3035:
3034:
3024:
3023:
3022:
2997:
2994:
2989:
2974:
2973:
2939:
2937:
2936:
2931:
2929:
2928:
2909:
2907:
2906:
2901:
2896:
2894:
2893:
2892:
2891:
2882:
2873:
2866:
2865:
2852:
2849:
2844:
2801:
2799:
2798:
2793:
2750:
2748:
2747:
2742:
2710:
2708:
2707:
2702:
2700:
2692:
2683:
2681:
2680:
2675:
2650:
2648:
2647:
2642:
2640:
2639:
2634:
2630:
2622:
2609:
2605:
2592:
2591:
2579:
2578:
2561:
2553:
2535:
2533:
2532:
2527:
2525:
2517:
2508:
2506:
2505:
2500:
2479:
2469:
2442:
2440:
2439:
2434:
2432:
2430:
2429:
2428:
2423:
2419:
2390:
2377:
2376:
2364:
2363:
2353:
2327:
2326:
2321:
2320:
2311:
2299:
2298:
2293:
2292:
2283:
2280:
2278:
2264:
2241:
2238:
2233:
2223:
2218:
2195:
2193:
2192:
2187:
2176:
2175:
2170:
2169:
2160:
2131:
2129:
2128:
2123:
2121:
2120:
2104:
2102:
2101:
2096:
2094:
2093:
2077:
2075:
2074:
2069:
2051:
2049:
2048:
2043:
2041:
2040:
2024:
2022:
2021:
2016:
1995:
1993:
1992:
1987:
1985:
1983:
1982:
1973:
1972:
1963:
1951:
1949:
1948:
1943:
1938:
1937:
1932:
1931:
1930:
1921:
1909:
1901:
1900:
1895:
1894:
1893:
1880:
1871:
1866:
1848:
1847:
1825:
1823:
1822:
1817:
1815:
1814:
1809:
1808:
1807:
1798:
1786:
1778:
1777:
1772:
1771:
1770:
1757:
1749:
1748:
1732:
1727:
1709:
1708:
1689:
1687:
1686:
1681:
1679:
1678:
1656:
1654:
1653:
1648:
1646:
1645:
1629:
1627:
1626:
1621:
1619:
1618:
1599:
1597:
1596:
1591:
1586:
1584:
1583:
1582:
1581:
1572:
1560:
1553:
1552:
1551:
1538:
1531:
1530:
1517:
1516:
1515:
1490:
1487:
1482:
1464:
1462:
1461:
1449:
1446:
1441:
1423:
1422:
1397:
1395:
1394:
1389:
1368:
1366:
1365:
1360:
1358:
1357:
1329:
1327:
1326:
1321:
1316:
1314:
1313:
1312:
1311:
1302:
1293:
1286:
1285:
1275:
1274:
1273:
1248:
1245:
1240:
1225:
1217:
1190:
1188:
1187:
1182:
1151:
1149:
1148:
1143:
1141:
1139:
1138:
1120:
1115:
1111:
1110:
1102:
1067:
1065:
1064:
1059:
1021:
1019:
1018:
1013:
992:
990:
989:
984:
955:
953:
952:
947:
922:
920:
919:
914:
909:
907:
906:
888:
865:
849:
847:
846:
841:
800:
798:
797:
792:
768:
766:
765:
760:
758:
757:
737:
735:
734:
729:
727:
726:
707:
705:
704:
699:
697:
695:
675:
674:
673:
652:
651:
638:
636:
635:
583:
581:
580:
575:
551:
549:
548:
543:
538:
520:
518:
517:
512:
510:
509:
497:
489:
478:
476:
475:
460:
455:
453:
452:
437:
432:
430:
429:
414:
409:
407:
406:
391:
360:
358:
357:
352:
334:
332:
331:
326:
293:
291:
290:
285:
267:
265:
264:
259:
217:
215:
214:
209:
204:
186:
184:
183:
178:
161:
159:
158:
146:
141:
139:
138:
126:
121:
119:
118:
106:
101:
99:
98:
86:
83:
78:
33:Apéry's constant
31:that states the
3842:
3841:
3837:
3836:
3835:
3833:
3832:
3831:
3812:
3811:
3797:10.2307/2695383
3780:
3775:
3772:
3767:
3719:
3718:
3714:
3684:
3683:
3679:
3631:
3630:
3626:
3617:
3613:
3583:
3577:10.4213/mzm1785
3560:
3555:
3554:
3550:
3534:
3533:
3529:
3507:
3506:
3502:
3492:
3491:
3487:
3469:
3468:
3464:
3430:
3425:
3424:
3420:
3389:
3388:
3384:
3380:
3334:
3333:
3305:
3304:
3276:
3275:
3247:
3246:
3209:
3208:
3180:
3175:
3174:
3145:
3140:
3139:
3111:
3110:
3089:
3084:
3083:
3044:
3037:
3026:
3025:
3008:
2998:
2965:
2945:
2944:
2920:
2915:
2914:
2875:
2868:
2857:
2856:
2807:
2806:
2775:
2774:
2761:
2724:
2723:
2720:Yuri Nesterenko
2686:
2685:
2660:
2659:
2617:
2616:
2583:
2570:
2569:
2565:
2541:
2540:
2511:
2510:
2482:
2481:
2471:
2461:
2458:
2451:
2403:
2399:
2398:
2391:
2368:
2355:
2354:
2312:
2284:
2265:
2242:
2205:
2204:
2161:
2153:
2152:
2138:
2112:
2107:
2106:
2085:
2080:
2079:
2054:
2053:
2032:
2027:
2026:
1998:
1997:
1974:
1964:
1957:
1956:
1911:
1904:
1902:
1875:
1873:
1839:
1834:
1833:
1788:
1781:
1779:
1752:
1750:
1734:
1700:
1695:
1694:
1664:
1659:
1658:
1637:
1632:
1631:
1610:
1605:
1604:
1562:
1555:
1533:
1522:
1518:
1501:
1491:
1453:
1408:
1403:
1402:
1371:
1370:
1343:
1338:
1337:
1295:
1288:
1277:
1276:
1259:
1249:
1196:
1195:
1164:
1163:
1155:for some fixed
1124:
1094:
1090:
1085:
1084:
1050:
1049:
1042:
1029:Hendrik Lenstra
995:
994:
966:
965:
932:
931:
889:
866:
859:
858:
814:
813:
771:
770:
749:
744:
743:
718:
713:
712:
676:
662:
640:
639:
621:
589:
588:
554:
553:
526:
525:
498:
464:
441:
418:
395:
385:
384:
376:proved that if
371:
337:
336:
308:
307:
270:
269:
247:
246:
192:
191:
150:
130:
110:
90:
44:
43:
27:is a result in
25:Apéry's theorem
17:
12:
11:
5:
3840:
3838:
3830:
3829:
3824:
3814:
3813:
3810:
3809:
3791:(3): 222–231.
3771:
3770:External links
3768:
3766:
3765:
3712:
3693:(4): 774–776.
3677:
3624:
3611:
3592:(6): 625–636.
3570:(6): 865–880.
3566:(in Russian).
3564:Матем. Заметки
3548:
3527:
3516:(3): 268–272.
3500:
3485:
3462:
3443:(4): 195–203.
3418:
3399:(3): 231–233.
3381:
3379:
3376:
3350:
3347:
3344:
3341:
3321:
3318:
3315:
3312:
3292:
3289:
3286:
3283:
3263:
3260:
3257:
3254:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3187:
3183:
3152:
3148:
3127:
3124:
3121:
3118:
3096:
3092:
3080:
3079:
3068:
3059:
3054:
3050:
3047:
3041:
3033:
3029:
3021:
3018:
3015:
3011:
3007:
3004:
3001:
2993:
2988:
2985:
2982:
2978:
2972:
2968:
2964:
2961:
2958:
2955:
2952:
2927:
2923:
2911:
2910:
2899:
2890:
2885:
2881:
2878:
2872:
2864:
2860:
2855:
2848:
2843:
2840:
2837:
2833:
2829:
2826:
2823:
2820:
2817:
2814:
2791:
2788:
2785:
2782:
2771:
2770:
2760:
2757:
2740:
2737:
2734:
2731:
2698:
2695:
2673:
2670:
2667:
2652:
2651:
2638:
2633:
2628:
2625:
2620:
2615:
2612:
2608:
2604:
2601:
2598:
2595:
2590:
2586:
2582:
2577:
2573:
2568:
2564:
2559:
2556:
2551:
2548:
2523:
2520:
2498:
2495:
2492:
2489:
2456:
2449:
2444:
2443:
2427:
2422:
2418:
2415:
2412:
2409:
2406:
2402:
2397:
2394:
2389:
2386:
2383:
2380:
2375:
2371:
2367:
2362:
2358:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2324:
2319:
2315:
2308:
2305:
2302:
2296:
2291:
2287:
2277:
2274:
2271:
2268:
2263:
2260:
2257:
2254:
2251:
2248:
2245:
2237:
2232:
2228:
2222:
2217:
2213:
2185:
2182:
2179:
2173:
2168:
2164:
2148:involving the
2137:
2134:
2119:
2115:
2092:
2088:
2067:
2064:
2061:
2039:
2035:
2014:
2011:
2008:
2005:
1981:
1977:
1971:
1967:
1953:
1952:
1941:
1936:
1929:
1924:
1920:
1917:
1914:
1908:
1899:
1892:
1887:
1884:
1879:
1870:
1865:
1862:
1859:
1855:
1851:
1846:
1842:
1827:
1826:
1813:
1806:
1801:
1797:
1794:
1791:
1785:
1776:
1769:
1764:
1761:
1756:
1747:
1744:
1741:
1737:
1731:
1726:
1723:
1720:
1716:
1712:
1707:
1703:
1677:
1674:
1671:
1667:
1644:
1640:
1617:
1613:
1601:
1600:
1589:
1580:
1575:
1571:
1568:
1565:
1559:
1550:
1545:
1542:
1537:
1529:
1525:
1521:
1514:
1511:
1508:
1504:
1500:
1497:
1494:
1486:
1481:
1478:
1475:
1471:
1467:
1460:
1456:
1452:
1445:
1440:
1437:
1434:
1430:
1426:
1421:
1418:
1415:
1411:
1387:
1384:
1381:
1378:
1356:
1353:
1350:
1346:
1331:
1330:
1319:
1310:
1305:
1301:
1298:
1292:
1284:
1280:
1272:
1269:
1266:
1262:
1258:
1255:
1252:
1244:
1239:
1236:
1233:
1229:
1223:
1220:
1215:
1212:
1209:
1206:
1203:
1180:
1177:
1174:
1171:
1153:
1152:
1137:
1134:
1131:
1127:
1123:
1118:
1114:
1108:
1105:
1100:
1097:
1093:
1057:
1041:
1038:
1011:
1008:
1005:
1002:
982:
979:
976:
973:
945:
942:
939:
928:transcendental
924:
923:
912:
905:
902:
899:
896:
892:
887:
884:
881:
878:
875:
872:
869:
839:
836:
833:
830:
827:
824:
821:
810:zeta constants
790:
787:
784:
781:
778:
756:
752:
725:
721:
709:
708:
694:
691:
688:
685:
682:
679:
672:
669:
665:
661:
658:
655:
650:
647:
643:
634:
631:
628:
624:
620:
617:
614:
611:
608:
605:
602:
599:
596:
573:
570:
567:
564:
561:
541:
537:
533:
522:
521:
508:
505:
501:
495:
492:
487:
484:
481:
474:
471:
467:
463:
458:
451:
448:
444:
440:
435:
428:
425:
421:
417:
412:
405:
402:
398:
394:
374:Leonhard Euler
370:
367:
350:
347:
344:
324:
321:
318:
315:
302:or not at the
283:
280:
277:
257:
254:
207:
203:
199:
188:
187:
176:
173:
170:
167:
164:
157:
153:
149:
144:
137:
133:
129:
124:
117:
113:
109:
104:
97:
93:
89:
82:
77:
74:
71:
67:
63:
60:
57:
54:
51:
15:
13:
10:
9:
6:
4:
3:
2:
3839:
3828:
3825:
3823:
3820:
3819:
3817:
3806:
3802:
3798:
3794:
3790:
3786:
3779:
3774:
3773:
3769:
3761:
3757:
3753:
3749:
3745:
3741:
3736:
3731:
3727:
3723:
3716:
3713:
3708:
3704:
3700:
3696:
3692:
3688:
3681:
3678:
3673:
3669:
3665:
3661:
3657:
3653:
3648:
3643:
3639:
3635:
3628:
3625:
3621:
3615:
3612:
3607:
3603:
3599:
3595:
3591:
3587:
3578:
3573:
3569:
3565:
3561:
3552:
3549:
3543:
3538:
3531:
3528:
3523:
3519:
3515:
3511:
3504:
3501:
3496:
3489:
3486:
3481:
3477:
3473:
3466:
3463:
3458:
3454:
3450:
3446:
3442:
3438:
3437:
3429:
3422:
3419:
3414:
3410:
3406:
3402:
3398:
3395:
3394:
3386:
3383:
3377:
3375:
3373:
3368:
3364:
3345:
3339:
3316:
3310:
3287:
3281:
3258:
3252:
3229:
3226:
3223:
3220:
3214:
3206:
3205:Wadim Zudilin
3201:
3185:
3181:
3172:
3168:
3150:
3146:
3122:
3116:
3094:
3090:
3066:
3052:
3048:
3045:
3031:
3027:
3019:
3016:
3013:
3005:
3002:
2986:
2983:
2980:
2976:
2970:
2966:
2962:
2956:
2950:
2943:
2942:
2941:
2925:
2921:
2897:
2883:
2879:
2876:
2862:
2858:
2853:
2841:
2838:
2835:
2831:
2827:
2824:
2818:
2812:
2805:
2804:
2803:
2786:
2780:
2769:
2768:
2763:
2762:
2758:
2756:
2754:
2735:
2729:
2721:
2717:
2716:Wadim Zudilin
2712:
2696:
2693:
2665:
2657:
2656:contradiction
2636:
2631:
2626:
2623:
2618:
2613:
2610:
2606:
2599:
2593:
2588:
2584:
2580:
2575:
2571:
2566:
2562:
2557:
2554:
2549:
2546:
2539:
2538:
2537:
2521:
2518:
2493:
2487:
2478:
2474:
2468:
2464:
2459:
2452:
2425:
2420:
2416:
2413:
2410:
2407:
2404:
2400:
2395:
2392:
2384:
2378:
2373:
2369:
2365:
2360:
2356:
2349:
2346:
2343:
2340:
2337:
2331:
2317:
2313:
2303:
2289:
2285:
2275:
2272:
2269:
2266:
2258:
2255:
2249:
2246:
2243:
2235:
2230:
2226:
2220:
2215:
2211:
2203:
2202:
2201:
2199:
2180:
2166:
2162:
2151:
2147:
2143:
2142:Frits Beukers
2135:
2133:
2117:
2113:
2090:
2086:
2065:
2062:
2059:
2037:
2033:
2009:
2003:
1996:converges to
1979:
1975:
1969:
1965:
1955:The sequence
1939:
1934:
1922:
1918:
1915:
1912:
1897:
1885:
1882:
1868:
1863:
1860:
1857:
1853:
1849:
1844:
1840:
1832:
1831:
1830:
1811:
1799:
1795:
1792:
1789:
1774:
1762:
1759:
1745:
1742:
1739:
1735:
1729:
1724:
1721:
1718:
1714:
1710:
1705:
1701:
1693:
1692:
1691:
1675:
1672:
1669:
1665:
1642:
1638:
1615:
1611:
1587:
1573:
1569:
1566:
1563:
1543:
1540:
1527:
1523:
1519:
1512:
1509:
1506:
1498:
1495:
1484:
1479:
1476:
1473:
1469:
1465:
1458:
1454:
1450:
1443:
1438:
1435:
1432:
1428:
1424:
1419:
1416:
1413:
1409:
1401:
1400:
1399:
1382:
1376:
1354:
1351:
1348:
1344:
1336:
1317:
1303:
1299:
1296:
1282:
1278:
1270:
1267:
1264:
1256:
1253:
1237:
1234:
1231:
1227:
1221:
1218:
1213:
1207:
1201:
1194:
1193:
1192:
1175:
1169:
1160:
1158:
1135:
1132:
1129:
1125:
1121:
1116:
1112:
1106:
1103:
1098:
1095:
1091:
1083:
1082:
1081:
1079:
1075:
1071:
1055:
1047:
1040:Apéry's proof
1037:
1034:
1030:
1026:
1006:
1000:
977:
971:
963:
957:
943:
940:
937:
929:
910:
903:
900:
897:
894:
890:
882:
879:
876:
873:
867:
857:
856:
855:
853:
834:
831:
828:
825:
819:
811:
806:
804:
785:
782:
776:
754:
750:
741:
723:
719:
692:
686:
683:
677:
670:
667:
659:
656:
648:
645:
641:
632:
629:
626:
618:
615:
609:
603:
600:
594:
587:
586:
585:
568:
565:
559:
539:
535:
531:
506:
503:
499:
493:
490:
485:
482:
479:
472:
469:
465:
461:
456:
449:
446:
442:
438:
433:
426:
423:
419:
415:
410:
403:
400:
396:
392:
383:
382:
381:
379:
375:
368:
366:
364:
348:
345:
342:
322:
319:
316:
313:
305:
301:
297:
281:
278:
275:
255:
252:
244:
240:
235:
233:
229:
225:
221:
205:
201:
197:
174:
171:
168:
165:
162:
155:
151:
147:
142:
135:
131:
127:
122:
115:
111:
107:
102:
95:
91:
87:
75:
72:
69:
65:
61:
55:
49:
42:
41:
40:
38:
34:
30:
29:number theory
26:
22:
3788:
3784:
3725:
3721:
3715:
3690:
3686:
3680:
3647:math/0008051
3637:
3633:
3627:
3619:
3614:
3589:
3585:
3567:
3563:
3551:
3542:math/0202159
3530:
3513:
3509:
3503:
3494:
3488:
3479:
3475:
3465:
3440:
3434:
3421:
3396:
3391:
3385:
3367:vector space
3202:
3081:
2912:
2772:
2764:
2713:
2653:
2454:
2447:
2445:
2139:
2136:Later proofs
1954:
1828:
1602:
1332:
1161:
1159:, δ > 0.
1156:
1154:
1077:
1073:
1043:
958:
925:
851:
807:
802:
710:
584:, he showed
523:
377:
372:
236:
223:
219:
189:
24:
18:
3640:: 267–270.
3586:Math. Notes
2654:which is a
2460:(sequences
1025:Henri Cohen
962:Roger Apéry
363:conjectured
232:Roger Apéry
21:mathematics
3816:Categories
3476:Astérisque
3378:References
3082:If such a
1080:such that
711:where the
37:irrational
3760:119143600
3672:119678120
3606:117487836
3457:121589323
3413:121346325
3363:dimension
3340:ζ
3311:ζ
3282:ζ
3253:ζ
3215:ζ
3147:ξ
3117:ζ
3091:ξ
3017:−
3003:−
2992:∞
2977:∑
2967:ξ
2951:ζ
2922:ξ
2847:∞
2832:∑
2813:ζ
2781:ζ
2765:See also
2730:ζ
2672:∞
2669:→
2611:≤
2594:ζ
2563:≤
2488:ζ
2411:…
2396:
2379:ζ
2323:~
2295:~
2270:−
2250:
2244:−
2227:∫
2212:∫
2172:~
2146:integrals
2004:ζ
1854:∑
1715:∑
1510:−
1496:−
1470:∑
1429:∑
1377:ζ
1268:−
1254:−
1243:∞
1228:∑
1202:ζ
1170:ζ
1136:δ
1099:−
1096:ξ
1072:integers
1056:ξ
1001:ζ
972:ζ
941:≥
891:π
868:ζ
820:ζ
777:ζ
751:π
660:π
616:−
595:ζ
560:ζ
500:π
483:⋯
306:integers
245:integers
175:…
172:1.2020569
166:⋯
81:∞
66:∑
50:ζ
3482:: 11–13.
3203:Work by
1335:sequence
300:rational
228:integers
35:ζ(3) is
3805:2695383
3740:Bibcode
3695:Bibcode
3652:Bibcode
3622:, 2007.
2477:A171485
2475::
2467:A171484
2465::
1070:coprime
1036:ideas.
369:History
3803:
3758:
3670:
3604:
3455:
3411:
3332:, and
1031:, and
218:where
3801:JSTOR
3781:(PDF)
3756:S2CID
3730:arXiv
3668:S2CID
3642:arXiv
3602:S2CID
3537:arXiv
3453:S2CID
3431:(PDF)
3409:S2CID
3365:of a
2550:<
2473:OEIS
2470:and
2463:OEIS
2453:and
2105:and
1829:and
1630:and
1117:<
1076:and
993:and
926:are
346:>
279:>
243:even
226:are
222:and
3793:doi
3789:108
3748:doi
3703:doi
3660:doi
3638:331
3594:doi
3572:doi
3518:doi
3445:doi
3401:doi
3186:383
2393:lcm
2247:log
1191:as
304:odd
241:at
19:In
3818::
3799:.
3787:.
3783:.
3754:.
3746:.
3738:.
3726:35
3724:.
3701:.
3691:56
3689:.
3666:.
3658:.
3650:.
3636:.
3600:.
3590:59
3588:.
3568:59
3562:.
3514:11
3512:.
3480:61
3478:.
3474:.
3451:.
3439:.
3433:.
3407:.
3397:99
3374:.
3346:11
3303:,
3274:,
3182:10
2755:.
2711:.
1027:,
956:.
805:.
234:.
23:,
3807:.
3795::
3762:.
3750::
3742::
3732::
3709:.
3705::
3697::
3674:.
3662::
3654::
3644::
3608:.
3596::
3580:.
3574::
3545:.
3539::
3524:.
3520::
3459:.
3447::
3441:1
3415:.
3403::
3349:)
3343:(
3320:)
3317:9
3314:(
3291:)
3288:7
3285:(
3262:)
3259:5
3256:(
3233:)
3230:1
3227:+
3224:n
3221:2
3218:(
3151:5
3126:)
3123:5
3120:(
3095:5
3067:.
3058:)
3053:n
3049:n
3046:2
3040:(
3032:5
3028:n
3020:1
3014:n
3010:)
3006:1
3000:(
2987:1
2984:=
2981:n
2971:5
2963:=
2960:)
2957:5
2954:(
2926:5
2898:.
2889:)
2884:n
2880:n
2877:2
2871:(
2863:2
2859:n
2854:1
2842:1
2839:=
2836:n
2828:3
2825:=
2822:)
2819:2
2816:(
2790:)
2787:2
2784:(
2739:)
2736:3
2733:(
2697:b
2694:1
2666:n
2637:n
2632:)
2627:5
2624:4
2619:(
2614:4
2607:|
2603:)
2600:3
2597:(
2589:n
2585:B
2581:+
2576:n
2572:A
2567:|
2558:b
2555:1
2547:0
2522:b
2519:a
2497:)
2494:3
2491:(
2457:n
2455:B
2450:n
2448:A
2426:3
2421:]
2417:n
2414:,
2408:,
2405:1
2401:[
2388:)
2385:3
2382:(
2374:n
2370:B
2366:+
2361:n
2357:A
2350:=
2347:y
2344:d
2341:x
2338:d
2335:)
2332:y
2329:(
2318:n
2314:P
2307:)
2304:x
2301:(
2290:n
2286:P
2276:y
2273:x
2267:1
2262:)
2259:y
2256:x
2253:(
2236:1
2231:0
2221:1
2216:0
2184:)
2181:x
2178:(
2167:n
2163:P
2118:n
2114:b
2091:n
2087:a
2066:2
2063:=
2060:n
2038:n
2034:a
2013:)
2010:3
2007:(
1980:n
1976:b
1970:n
1966:a
1940:.
1935:2
1928:)
1923:k
1919:k
1916:+
1913:n
1907:(
1898:2
1891:)
1886:k
1883:n
1878:(
1869:n
1864:0
1861:=
1858:k
1850:=
1845:n
1841:b
1812:2
1805:)
1800:k
1796:k
1793:+
1790:n
1784:(
1775:2
1768:)
1763:k
1760:n
1755:(
1746:k
1743:,
1740:n
1736:c
1730:n
1725:0
1722:=
1719:k
1711:=
1706:n
1702:a
1676:k
1673:,
1670:n
1666:c
1643:n
1639:b
1616:n
1612:a
1588:.
1579:)
1574:m
1570:m
1567:+
1564:n
1558:(
1549:)
1544:m
1541:n
1536:(
1528:3
1524:m
1520:2
1513:1
1507:m
1503:)
1499:1
1493:(
1485:k
1480:1
1477:=
1474:m
1466:+
1459:3
1455:m
1451:1
1444:n
1439:1
1436:=
1433:m
1425:=
1420:k
1417:,
1414:n
1410:c
1386:)
1383:3
1380:(
1355:k
1352:,
1349:n
1345:c
1318:.
1309:)
1304:n
1300:n
1297:2
1291:(
1283:3
1279:n
1271:1
1265:n
1261:)
1257:1
1251:(
1238:1
1235:=
1232:n
1222:2
1219:5
1214:=
1211:)
1208:3
1205:(
1179:)
1176:3
1173:(
1157:c
1133:+
1130:1
1126:q
1122:c
1113:|
1107:q
1104:p
1092:|
1078:q
1074:p
1010:)
1007:2
1004:(
981:)
978:3
975:(
944:1
938:n
911:,
904:1
901:+
898:n
895:2
886:)
883:1
880:+
877:n
874:2
871:(
852:n
838:)
835:1
832:+
829:n
826:2
823:(
803:n
789:)
786:n
783:2
780:(
755:n
724:n
720:B
693:!
690:)
687:n
684:2
681:(
678:2
671:n
668:2
664:)
657:2
654:(
649:n
646:2
642:B
633:1
630:+
627:n
623:)
619:1
613:(
610:=
607:)
604:n
601:2
598:(
572:)
569:n
566:2
563:(
540:q
536:/
532:p
507:n
504:2
494:q
491:p
486:=
480:+
473:n
470:2
466:4
462:1
457:+
450:n
447:2
443:3
439:1
434:+
427:n
424:2
420:2
416:1
411:+
404:n
401:2
397:1
393:1
378:n
349:1
343:n
335:(
323:1
320:+
317:n
314:2
282:0
276:n
268:(
256:n
253:2
224:q
220:p
206:q
202:/
198:p
169:=
163:+
156:3
152:3
148:1
143:+
136:3
132:2
128:1
123:+
116:3
112:1
108:1
103:=
96:3
92:n
88:1
76:1
73:=
70:n
62:=
59:)
56:3
53:(
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