2997:
2551:
2992:{\displaystyle {\begin{aligned}\alpha _{k}&\leq {\frac {3k-4}{4k}}\quad (4\leq k\leq 8)\\\alpha _{9}&\leq {\frac {35}{54}}\ ,\quad \alpha _{10}\leq {\frac {41}{60}}\ ,\quad \alpha _{11}\leq {\frac {7}{10}}\\\alpha _{k}&\leq {\frac {k-2}{k+2}}\quad (12\leq k\leq 25)\\\alpha _{k}&\leq {\frac {k-1}{k+4}}\quad (26\leq k\leq 50)\\\alpha _{k}&\leq {\frac {31k-98}{32k}}\quad (51\leq k\leq 57)\\\alpha _{k}&\leq {\frac {7k-34}{7k}}\quad (k\geq 58)\end{aligned}}}
71:
31:
111:
764:
3409:
3650:
3204:
528:
2556:
2365:
640:
280:
2186:
1125:
1728:
1512:
3274:
377:
1672:
1626:
1842:
1792:
2026:
802:
1566:
921:
2487:
1004:
2539:
3063:
1456:
834:
Finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible to give approximations. The leading behavior of the series is given by
1572:
1406:
1346:
3522:
3451:
1269:
4018:
G. Kolesnik. On the estimation of multiple exponential sums, in "Recent
Progress in Analytic Number Theory", Symposium Durham 1979 (Vol. 1), Academic, London, 1981, pp. 231–246.
1921:
3083:
2391:
1226:
3689:
1154:
632:
1869:
2418:
2292:
581:) counts the number of points on a square lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis, and to the upper-right by the hyperbola
145:
105:
65:
3262:
3715:
2438:
2215:
1063:
944:
3233:
2056:
3480:
3506:
2241:
2085:
2261:
1289:
1027:
396:
1871:
lies somewhere between 1/4 and 131/416 (approx. 0.3149); it is widely conjectured to be 1/4. Theoretical evidence lends credence to this conjecture, since
3839:
1738:
1518:
2297:
759:{\displaystyle D(x)=\sum _{k=1}^{x}\left\lfloor {\frac {x}{k}}\right\rfloor =2\sum _{k=1}^{u}\left\lfloor {\frac {x}{k}}\right\rfloor -u^{2}}
185:
2093:
147:, graphed as a distribution or histogram. The vertical scale is not constant left to right; click on image for a detailed description.
4147:
821:
17:
3404:{\displaystyle \Delta (x)={\frac {1}{2\pi i}}\int _{c^{\prime }-i\infty }^{c^{\prime }+i\infty }\zeta ^{2}(w){\frac {x^{w}}{w}}\,dw}
1071:
1680:
1464:
291:
1634:
1578:
3741:
1804:
1744:
1941:
1156:. As of today, this problem remains unsolved. Progress has been slow. Many of the same methods work for this problem and for
826:
0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, ...
807:
If the hyperbola in this context is replaced by a circle then determining the value of the resulting function is known as the
769:
4083:
3781:
3753:
1528:
840:
554:
numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in
2446:
956:
2498:
3009:
4055:
3995:
3819:
1418:
3979:
3737:
4043:
1038:
3645:{\displaystyle D_{k}(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\zeta ^{k}(w){\frac {x^{w}}{w}}\,dw}
4137:
3987:
1355:
947:
1298:
3513:
3417:
2263:. Computing these infima is known as the Piltz divisor problem, after the name of the German mathematician
3199:{\displaystyle D(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\zeta ^{2}(w){\frac {x^{w}}{w}}\,dw}
1237:
1874:
3509:
2370:
1175:
1034:
4030:. The Theory of the Riemann Zeta-function with Applications (Theorem 13.2). John Wiley and Sons 1985.
1923:
has a (non-Gaussian) limiting distribution. The value of 1/4 would also follow from a conjecture on
1157:
3658:
1133:
604:
1851:
4142:
2396:
2270:
117:
77:
37:
3238:
2059:
3694:
3265:
2423:
2194:
1048:
929:
164:
3212:
4093:
2034:
808:
4101:
4005:
3910:
3456:
523:{\displaystyle D_{k}(x)=\sum _{n\leq x}d_{k}(n)=\sum _{m\leq x}\sum _{mn\leq x}d_{k-1}(n)}
8:
3485:
2220:
2064:
3958:
2246:
1274:
1012:
4027:
4079:
4051:
3991:
3950:
3898:
3860:
3843:
3815:
3777:
3749:
1412:
3962:
1045:, precisely stated, is to improve this error bound by finding the smallest value of
167:. The various studies of the behaviour of the divisor function are sometimes called
4097:
4001:
3940:
3906:
3890:
3855:
3074:
3003:
383:
160:
3984:
Ten lectures on the interface between analytic number theory and harmonic analysis
4089:
3929:"The distribution and moments of the error term in the Dirichlet divisor problem"
4071:
3835:
3769:
1924:
1030:
3894:
4131:
4117:
3954:
3902:
3878:
1798:
1522:
152:
4078:, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag,
3945:
3928:
3986:. Regional Conference Series in Mathematics. Vol. 84. Providence, RI:
3924:
2360:{\displaystyle \Delta _{k}(x)=O\left(x^{\alpha _{k}+\varepsilon }\right)}
2264:
1231:
390:
can be written as a product of two integers. More generally, one defines
110:
70:
1734:
1169:
163:. It frequently occurs in the study of the asymptotic behaviour of the
4106:(Provides an introductory statement of the Dirichlet divisor problem.)
4066:(See chapter 12 for a discussion of the generalized divisor problem)
275:{\displaystyle D(x)=\sum _{n\leq x}d(n)=\sum _{j,k \atop jk\leq x}1}
30:
2181:{\displaystyle \Delta _{k}(x)=O\left(x^{1-1/k}\log ^{k-2}x\right)}
386:. The divisor function counts the number of ways that the integer
590:
1120:{\displaystyle \Delta (x)=O\left(x^{\theta +\epsilon }\right)}
2243:
case, the infimum of the bound is not known for any value of
1723:{\displaystyle \inf \theta \leq 35/108=0.32{\overline {407}}}
1507:{\displaystyle \inf \theta \leq 27/82=0.3{\overline {29268}}}
3482:
is obtained by shifting the contour past the double pole at
816:
372:{\displaystyle d(n)=\sigma _{0}(n)=\sum _{j,k \atop jk=n}1}
1667:{\displaystyle \inf \theta \leq 346/1067=0.32427366448...}
1621:{\displaystyle \inf \theta \leq 12/37=0.{\overline {324}}}
1164:
surveys what is known and not known about these problems.
593:. This allows us to provide an alternative expression for
1837:{\displaystyle \inf \theta \leq 131/416=0.31490384615...}
1787:{\displaystyle \inf \theta \leq 7/22=0.3{\overline {18}}}
1160:, another lattice-point counting problem. Section F1 of
2021:{\displaystyle D_{k}(x)=xP_{k}(\log x)+\Delta _{k}(x)\,}
1271:. In particular, he demonstrated that for some constant
797:{\displaystyle u=\left\lfloor {\sqrt {x}}\right\rfloor }
589:. Roughly, this shape may be envisioned as a hyperbolic
114:
The summatory function, with leading terms removed, for
74:
The summatory function, with leading terms removed, for
34:
The summatory function, with leading terms removed, for
3736:
1561:{\displaystyle \inf \theta \leq 15/46=0.32608695652...}
916:{\displaystyle D(x)=x\log x+x(2\gamma -1)+\Delta (x)\ }
3834:
3697:
3661:
3525:
3488:
3459:
3420:
3277:
3241:
3215:
3086:
3012:
2554:
2501:
2482:{\displaystyle \alpha _{2}\leq {\frac {131}{416}}\ ,}
2449:
2426:
2399:
2373:
2300:
2273:
2249:
2223:
2197:
2096:
2067:
2037:
1944:
1877:
1854:
1807:
1747:
1683:
1637:
1581:
1531:
1467:
1421:
1358:
1301:
1277:
1240:
1178:
1136:
1074:
1051:
1015:
999:{\displaystyle \Delta (x)=O\left({\sqrt {x}}\right).}
959:
932:
843:
772:
643:
607:
399:
294:
188:
120:
80:
40:
3732:
3730:
4120:(2003) 'Exponential Sums and Lattice Points III',
3881:(2003). "Exponential sums and lattice points III".
2534:{\displaystyle \alpha _{3}\leq {\frac {43}{96}}\ ,}
2087:. Using simple estimates, it is readily shown that
27:
Summatory function of the divisor-counting function
3967:Theorem 1 The function has a distribution function
3709:
3683:
3644:
3500:
3474:
3445:
3403:
3256:
3227:
3198:
3057:
2991:
2533:
2481:
2432:
2412:
2385:
2359:
2286:
2255:
2235:
2209:
2180:
2079:
2050:
2020:
1915:
1863:
1836:
1786:
1722:
1666:
1620:
1560:
1506:
1450:
1400:
1340:
1283:
1263:
1220:
1148:
1119:
1057:
1021:
998:
938:
915:
796:
758:
626:
522:
371:
274:
139:
99:
59:
3727:
4129:
4064:, (1951) Oxford at the Clarendon Press, Oxford.
3746:Multiplicative Number Theory I: Classical Theory
3058:{\displaystyle \alpha _{k}={\frac {k-1}{2k}}\ .}
2267:(also see his German page). Defining the order
1855:
1808:
1748:
1684:
1638:
1582:
1532:
1468:
1422:
1241:
1172:proved that the error term can be improved to
829:
179:The divisor summatory function is defined as
1451:{\displaystyle \inf \theta \leq 33/100=0.33}
3923:
2393:, one has the following results (note that
3978:
3944:
3859:
3748:. Cambridge: Cambridge University Press.
3635:
3394:
3189:
2017:
3873:
3871:
1930:
1033:. This estimate can be proven using the
109:
69:
29:
4070:
4062:The theory of the Riemann Zeta-Function
1401:{\displaystyle \Delta (x)<-Kx^{1/4}}
14:
4130:
4076:Introduction to analytic number theory
3877:
3805:
3803:
3801:
3799:
3797:
3795:
3793:
1341:{\displaystyle \Delta (x)>Kx^{1/4}}
3868:
3446:{\displaystyle 0<c^{\prime }<1}
601:), and a simple way to compute it in
159:is a function that is a sum over the
3972:
3844:"On the divisor and circle problems"
3809:
3762:
1677:In 1982, Kolesnik demonstrated that
1631:In 1973, Kolesnik demonstrated that
1461:In 1928, van der Corput proved that
1264:{\displaystyle \inf \theta \geq 1/4}
3790:
3768:
3068:
24:
3776:(3rd ed.). Berlin: Springer.
3774:Unsolved Problems in Number Theory
3663:
3594:
3580:
3432:
3353:
3342:
3332:
3321:
3278:
3148:
3134:
3073:Both portions may be expressed as
2302:
2098:
1999:
1916:{\displaystyle \Delta (x)/x^{1/4}}
1878:
1359:
1302:
1162:Unsolved Problems in Number Theory
1075:
960:
898:
337:
240:
25:
4159:
2386:{\displaystyle \varepsilon >0}
1935:In the generalized case, one has
1221:{\displaystyle O(x^{1/3}\log x).}
546:) counts the number of ways that
4148:Unsolved problems in mathematics
3814:. New York: Dover Publications.
2294:as the smallest value for which
3508:: the leading term is just the
2969:
2900:
2831:
2762:
2691:
2661:
2602:
1037:, and was first established by
550:can be written as a product of
4021:
4012:
3917:
3828:
3684:{\displaystyle \Delta _{k}(x)}
3678:
3672:
3615:
3609:
3542:
3536:
3469:
3463:
3374:
3368:
3287:
3281:
3251:
3245:
3169:
3163:
3096:
3090:
2982:
2970:
2919:
2901:
2850:
2832:
2781:
2763:
2621:
2603:
2317:
2311:
2113:
2107:
2014:
2008:
1992:
1980:
1961:
1955:
1887:
1881:
1415:improved Dirichlet's bound to
1368:
1362:
1311:
1305:
1212:
1182:
1149:{\displaystyle \epsilon >0}
1084:
1078:
969:
963:
907:
901:
892:
877:
853:
847:
653:
647:
627:{\displaystyle O({\sqrt {x}})}
621:
611:
517:
511:
454:
448:
416:
410:
326:
320:
304:
298:
229:
223:
198:
192:
13:
1:
4050:, (1974) Dover Publications,
4037:
3988:American Mathematical Society
174:
3861:10.1016/0022-314X(88)90093-5
1864:{\displaystyle \inf \theta }
1779:
1715:
1613:
1499:
7:
2413:{\displaystyle \alpha _{2}}
2287:{\displaystyle \alpha _{k}}
1801:improved this to show that
830:Dirichlet's divisor problem
140:{\displaystyle x<10^{7}}
100:{\displaystyle x<10^{7}}
60:{\displaystyle x<10^{4}}
10:
4164:
4112:A Course in Number Theory.
2440:of the previous section):
1521:and independently in 1953
1035:Dirichlet hyperbola method
814:Sequence of D(n)(sequence
157:divisor summatory function
3895:10.1112/S0024611503014485
3812:The Riemann Zeta-Function
3810:Ivic, Aleksandar (2003).
3514:Cauchy's integral formula
3257:{\displaystyle \zeta (s)}
1043:Dirichlet divisor problem
948:Euler–Mascheroni constant
18:Dirichlet divisor problem
3848:Journal of Number Theory
3720:
1291:, there exist values of
950:, and the error term is
4122:Proc. London Math. Soc.
4048:Riemann's Zeta Function
3946:10.4064/aa-60-4-389-415
3710:{\displaystyle k\geq 2}
3516:. In general, one has
2433:{\displaystyle \theta }
2210:{\displaystyle k\geq 2}
1058:{\displaystyle \theta }
939:{\displaystyle \gamma }
558:dimensions. Thus, for
3883:Proc. London Math. Soc
3711:
3685:
3646:
3502:
3476:
3453:. The leading term of
3447:
3405:
3258:
3229:
3228:{\displaystyle c>1}
3200:
3059:
2993:
2535:
2483:
2434:
2414:
2387:
2361:
2288:
2257:
2237:
2211:
2182:
2081:
2052:
2022:
1917:
1865:
1838:
1788:
1724:
1668:
1622:
1562:
1508:
1452:
1402:
1342:
1285:
1265:
1222:
1158:Gauss's circle problem
1150:
1121:
1059:
1023:
1000:
940:
917:
798:
760:
724:
679:
628:
524:
373:
276:
148:
141:
107:
101:
67:
61:
3712:
3686:
3647:
3503:
3477:
3448:
3406:
3268:. Similarly, one has
3266:Riemann zeta function
3259:
3230:
3201:
3060:
2994:
2536:
2484:
2435:
2415:
2388:
2362:
2289:
2258:
2238:
2212:
2183:
2082:
2053:
2051:{\displaystyle P_{k}}
2023:
1931:Piltz divisor problem
1918:
1866:
1839:
1789:
1725:
1669:
1623:
1563:
1509:
1453:
1403:
1343:
1286:
1266:
1223:
1151:
1122:
1060:
1024:
1001:
941:
918:
799:
761:
704:
659:
629:
525:
374:
277:
165:Riemann zeta function
142:
113:
102:
73:
62:
33:
4138:Arithmetic functions
3695:
3659:
3523:
3486:
3475:{\displaystyle D(x)}
3457:
3418:
3275:
3239:
3213:
3084:
3010:
2552:
2499:
2447:
2424:
2397:
2371:
2298:
2271:
2247:
2221:
2195:
2094:
2065:
2060:polynomial of degree
2035:
1942:
1875:
1852:
1805:
1745:
1681:
1635:
1579:
1529:
1465:
1419:
1356:
1299:
1275:
1238:
1176:
1134:
1072:
1049:
1013:
957:
930:
841:
809:Gauss circle problem
770:
641:
605:
397:
292:
186:
118:
78:
38:
3980:Montgomery, Hugh L.
3598:
3501:{\displaystyle w=1}
3357:
3152:
2236:{\displaystyle k=2}
2080:{\displaystyle k-1}
1130:holds true for all
4060:E. C. Titchmarsh,
3925:Heath-Brown, D. R.
3707:
3681:
3642:
3566:
3498:
3472:
3443:
3401:
3311:
3254:
3225:
3196:
3120:
3055:
2989:
2987:
2531:
2479:
2430:
2410:
2383:
2357:
2284:
2253:
2233:
2207:
2178:
2077:
2048:
2018:
1913:
1861:
1834:
1784:
1720:
1664:
1618:
1575:demonstrated that
1558:
1504:
1448:
1398:
1338:
1281:
1261:
1218:
1146:
1117:
1055:
1019:
996:
936:
913:
794:
756:
624:
520:
494:
475:
437:
369:
365:
272:
268:
219:
149:
137:
108:
97:
68:
57:
4085:978-0-387-90163-3
3783:978-0-387-20860-2
3755:978-0-521-84903-6
3655:and likewise for
3633:
3564:
3392:
3309:
3187:
3118:
3075:Mellin transforms
3051:
3047:
3006:conjectures that
2967:
2898:
2829:
2760:
2713:
2687:
2683:
2657:
2653:
2600:
2527:
2523:
2475:
2471:
2256:{\displaystyle k}
1782:
1718:
1616:
1502:
1413:J. van der Corput
1284:{\displaystyle K}
1022:{\displaystyle O}
987:
912:
788:
737:
692:
619:
476:
460:
422:
363:
332:
266:
235:
204:
16:(Redirected from
4155:
4104:
4031:
4025:
4019:
4016:
4010:
4009:
3976:
3970:
3969:
3948:
3933:Acta Arithmetica
3921:
3915:
3914:
3875:
3866:
3865:
3863:
3832:
3826:
3825:
3807:
3788:
3787:
3766:
3760:
3759:
3738:Montgomery, Hugh
3734:
3716:
3714:
3713:
3708:
3690:
3688:
3687:
3682:
3671:
3670:
3651:
3649:
3648:
3643:
3634:
3629:
3628:
3619:
3608:
3607:
3597:
3583:
3565:
3563:
3549:
3535:
3534:
3507:
3505:
3504:
3499:
3481:
3479:
3478:
3473:
3452:
3450:
3449:
3444:
3436:
3435:
3410:
3408:
3407:
3402:
3393:
3388:
3387:
3378:
3367:
3366:
3356:
3346:
3345:
3335:
3325:
3324:
3310:
3308:
3294:
3263:
3261:
3260:
3255:
3234:
3232:
3231:
3226:
3205:
3203:
3202:
3197:
3188:
3183:
3182:
3173:
3162:
3161:
3151:
3137:
3119:
3117:
3103:
3069:Mellin transform
3064:
3062:
3061:
3056:
3049:
3048:
3046:
3038:
3027:
3022:
3021:
3004:E. C. Titchmarsh
2998:
2996:
2995:
2990:
2988:
2968:
2966:
2958:
2944:
2935:
2934:
2899:
2897:
2889:
2875:
2866:
2865:
2830:
2828:
2817:
2806:
2797:
2796:
2761:
2759:
2748:
2737:
2728:
2727:
2714:
2706:
2701:
2700:
2685:
2684:
2676:
2671:
2670:
2655:
2654:
2646:
2637:
2636:
2601:
2599:
2591:
2577:
2568:
2567:
2540:
2538:
2537:
2532:
2525:
2524:
2516:
2511:
2510:
2488:
2486:
2485:
2480:
2473:
2472:
2464:
2459:
2458:
2439:
2437:
2436:
2431:
2419:
2417:
2416:
2411:
2409:
2408:
2392:
2390:
2389:
2384:
2366:
2364:
2363:
2358:
2356:
2352:
2351:
2344:
2343:
2310:
2309:
2293:
2291:
2290:
2285:
2283:
2282:
2262:
2260:
2259:
2254:
2242:
2240:
2239:
2234:
2216:
2214:
2213:
2208:
2187:
2185:
2184:
2179:
2177:
2173:
2166:
2165:
2150:
2149:
2145:
2106:
2105:
2086:
2084:
2083:
2078:
2057:
2055:
2054:
2049:
2047:
2046:
2027:
2025:
2024:
2019:
2007:
2006:
1979:
1978:
1954:
1953:
1922:
1920:
1919:
1914:
1912:
1911:
1907:
1894:
1870:
1868:
1867:
1862:
1843:
1841:
1840:
1835:
1832:0.31490384615...
1824:
1793:
1791:
1790:
1785:
1783:
1775:
1764:
1729:
1727:
1726:
1721:
1719:
1711:
1700:
1673:
1671:
1670:
1665:
1662:0.32427366448...
1654:
1627:
1625:
1624:
1619:
1617:
1609:
1598:
1573:Grigori Kolesnik
1567:
1565:
1564:
1559:
1556:0.32608695652...
1548:
1513:
1511:
1510:
1505:
1503:
1495:
1484:
1457:
1455:
1454:
1449:
1438:
1407:
1405:
1404:
1399:
1397:
1396:
1392:
1347:
1345:
1344:
1339:
1337:
1336:
1332:
1290:
1288:
1287:
1282:
1270:
1268:
1267:
1262:
1257:
1227:
1225:
1224:
1219:
1202:
1201:
1197:
1155:
1153:
1152:
1147:
1126:
1124:
1123:
1118:
1116:
1112:
1111:
1064:
1062:
1061:
1056:
1028:
1026:
1025:
1020:
1005:
1003:
1002:
997:
992:
988:
983:
945:
943:
942:
937:
922:
920:
919:
914:
910:
819:
803:
801:
800:
795:
793:
789:
784:
765:
763:
762:
757:
755:
754:
742:
738:
730:
723:
718:
697:
693:
685:
678:
673:
633:
631:
630:
625:
620:
615:
529:
527:
526:
521:
510:
509:
493:
474:
447:
446:
436:
409:
408:
384:divisor function
378:
376:
375:
370:
364:
362:
348:
319:
318:
281:
279:
278:
273:
267:
265:
251:
218:
169:divisor problems
161:divisor function
146:
144:
143:
138:
136:
135:
106:
104:
103:
98:
96:
95:
66:
64:
63:
58:
56:
55:
21:
4163:
4162:
4158:
4157:
4156:
4154:
4153:
4152:
4128:
4127:
4114:, Oxford, 1988.
4086:
4072:Apostol, Tom M.
4040:
4035:
4034:
4028:Aleksandar Ivić
4026:
4022:
4017:
4013:
3998:
3977:
3973:
3922:
3918:
3876:
3869:
3833:
3829:
3822:
3808:
3791:
3784:
3770:Guy, Richard K.
3767:
3763:
3756:
3735:
3728:
3723:
3696:
3693:
3692:
3666:
3662:
3660:
3657:
3656:
3624:
3620:
3618:
3603:
3599:
3584:
3570:
3553:
3548:
3530:
3526:
3524:
3521:
3520:
3487:
3484:
3483:
3458:
3455:
3454:
3431:
3427:
3419:
3416:
3415:
3383:
3379:
3377:
3362:
3358:
3341:
3337:
3336:
3320:
3316:
3315:
3298:
3293:
3276:
3273:
3272:
3240:
3237:
3236:
3214:
3211:
3210:
3178:
3174:
3172:
3157:
3153:
3138:
3124:
3107:
3102:
3085:
3082:
3081:
3071:
3039:
3028:
3026:
3017:
3013:
3011:
3008:
3007:
2986:
2985:
2959:
2945:
2943:
2936:
2930:
2926:
2923:
2922:
2890:
2876:
2874:
2867:
2861:
2857:
2854:
2853:
2818:
2807:
2805:
2798:
2792:
2788:
2785:
2784:
2749:
2738:
2736:
2729:
2723:
2719:
2716:
2715:
2705:
2696:
2692:
2675:
2666:
2662:
2645:
2638:
2632:
2628:
2625:
2624:
2592:
2578:
2576:
2569:
2563:
2559:
2555:
2553:
2550:
2549:
2545:
2515:
2506:
2502:
2500:
2497:
2496:
2492:
2463:
2454:
2450:
2448:
2445:
2444:
2425:
2422:
2421:
2404:
2400:
2398:
2395:
2394:
2372:
2369:
2368:
2367:holds, for any
2339:
2335:
2334:
2330:
2326:
2305:
2301:
2299:
2296:
2295:
2278:
2274:
2272:
2269:
2268:
2248:
2245:
2244:
2222:
2219:
2218:
2196:
2193:
2192:
2155:
2151:
2141:
2131:
2127:
2126:
2122:
2101:
2097:
2095:
2092:
2091:
2066:
2063:
2062:
2042:
2038:
2036:
2033:
2032:
2002:
1998:
1974:
1970:
1949:
1945:
1943:
1940:
1939:
1933:
1903:
1899:
1895:
1890:
1876:
1873:
1872:
1853:
1850:
1849:
1820:
1806:
1803:
1802:
1774:
1760:
1746:
1743:
1742:
1710:
1696:
1682:
1679:
1678:
1650:
1636:
1633:
1632:
1608:
1594:
1580:
1577:
1576:
1544:
1530:
1527:
1526:
1494:
1480:
1466:
1463:
1462:
1434:
1420:
1417:
1416:
1388:
1384:
1380:
1357:
1354:
1353:
1328:
1324:
1320:
1300:
1297:
1296:
1276:
1273:
1272:
1253:
1239:
1236:
1235:
1193:
1189:
1185:
1177:
1174:
1173:
1135:
1132:
1131:
1101:
1097:
1093:
1073:
1070:
1069:
1050:
1047:
1046:
1014:
1011:
1010:
982:
978:
958:
955:
954:
931:
928:
927:
842:
839:
838:
832:
825:
815:
783:
779:
771:
768:
767:
750:
746:
729:
725:
719:
708:
684:
680:
674:
663:
642:
639:
638:
614:
606:
603:
602:
576:
541:
499:
495:
480:
464:
442:
438:
426:
404:
400:
398:
395:
394:
349:
338:
336:
314:
310:
293:
290:
289:
252:
241:
239:
208:
187:
184:
183:
177:
131:
127:
119:
116:
115:
91:
87:
79:
76:
75:
51:
47:
39:
36:
35:
28:
23:
22:
15:
12:
11:
5:
4161:
4151:
4150:
4145:
4143:Lattice points
4140:
4126:
4125:
4124:(3)87: 591–609
4115:
4108:
4084:
4068:
4058:
4039:
4036:
4033:
4032:
4020:
4011:
3996:
3990:. p. 59.
3971:
3939:(4): 389–415.
3916:
3889:(3): 591–609.
3867:
3840:C. J. Mozzochi
3827:
3820:
3789:
3782:
3761:
3754:
3725:
3724:
3722:
3719:
3706:
3703:
3700:
3680:
3677:
3674:
3669:
3665:
3653:
3652:
3641:
3638:
3632:
3627:
3623:
3617:
3614:
3611:
3606:
3602:
3596:
3593:
3590:
3587:
3582:
3579:
3576:
3573:
3569:
3562:
3559:
3556:
3552:
3547:
3544:
3541:
3538:
3533:
3529:
3497:
3494:
3491:
3471:
3468:
3465:
3462:
3442:
3439:
3434:
3430:
3426:
3423:
3412:
3411:
3400:
3397:
3391:
3386:
3382:
3376:
3373:
3370:
3365:
3361:
3355:
3352:
3349:
3344:
3340:
3334:
3331:
3328:
3323:
3319:
3314:
3307:
3304:
3301:
3297:
3292:
3289:
3286:
3283:
3280:
3253:
3250:
3247:
3244:
3224:
3221:
3218:
3207:
3206:
3195:
3192:
3186:
3181:
3177:
3171:
3168:
3165:
3160:
3156:
3150:
3147:
3144:
3141:
3136:
3133:
3130:
3127:
3123:
3116:
3113:
3110:
3106:
3101:
3098:
3095:
3092:
3089:
3070:
3067:
3066:
3065:
3054:
3045:
3042:
3037:
3034:
3031:
3025:
3020:
3016:
3000:
2999:
2984:
2981:
2978:
2975:
2972:
2965:
2962:
2957:
2954:
2951:
2948:
2942:
2939:
2937:
2933:
2929:
2925:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2896:
2893:
2888:
2885:
2882:
2879:
2873:
2870:
2868:
2864:
2860:
2856:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2827:
2824:
2821:
2816:
2813:
2810:
2804:
2801:
2799:
2795:
2791:
2787:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2758:
2755:
2752:
2747:
2744:
2741:
2735:
2732:
2730:
2726:
2722:
2718:
2717:
2712:
2709:
2704:
2699:
2695:
2690:
2682:
2679:
2674:
2669:
2665:
2660:
2652:
2649:
2644:
2641:
2639:
2635:
2631:
2627:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2598:
2595:
2590:
2587:
2584:
2581:
2575:
2572:
2570:
2566:
2562:
2558:
2557:
2543:
2542:
2530:
2522:
2519:
2514:
2509:
2505:
2490:
2489:
2478:
2470:
2467:
2462:
2457:
2453:
2429:
2407:
2403:
2382:
2379:
2376:
2355:
2350:
2347:
2342:
2338:
2333:
2329:
2325:
2322:
2319:
2316:
2313:
2308:
2304:
2281:
2277:
2252:
2232:
2229:
2226:
2206:
2203:
2200:
2189:
2188:
2176:
2172:
2169:
2164:
2161:
2158:
2154:
2148:
2144:
2140:
2137:
2134:
2130:
2125:
2121:
2118:
2115:
2112:
2109:
2104:
2100:
2076:
2073:
2070:
2045:
2041:
2029:
2028:
2016:
2013:
2010:
2005:
2001:
1997:
1994:
1991:
1988:
1985:
1982:
1977:
1973:
1969:
1966:
1963:
1960:
1957:
1952:
1948:
1932:
1929:
1925:exponent pairs
1910:
1906:
1902:
1898:
1893:
1889:
1886:
1883:
1880:
1860:
1857:
1846:
1845:
1833:
1830:
1827:
1823:
1819:
1816:
1813:
1810:
1795:
1781:
1778:
1773:
1770:
1767:
1763:
1759:
1756:
1753:
1750:
1739:C. J. Mozzochi
1731:
1717:
1714:
1709:
1706:
1703:
1699:
1695:
1692:
1689:
1686:
1675:
1663:
1660:
1657:
1653:
1649:
1646:
1643:
1640:
1629:
1615:
1612:
1607:
1604:
1601:
1597:
1593:
1590:
1587:
1584:
1569:
1557:
1554:
1551:
1547:
1543:
1540:
1537:
1534:
1519:Chih Tsung-tao
1515:
1501:
1498:
1493:
1490:
1487:
1483:
1479:
1476:
1473:
1470:
1459:
1447:
1444:
1441:
1437:
1433:
1430:
1427:
1424:
1409:
1395:
1391:
1387:
1383:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1348:and values of
1335:
1331:
1327:
1323:
1319:
1316:
1313:
1310:
1307:
1304:
1280:
1260:
1256:
1252:
1249:
1246:
1243:
1228:
1217:
1214:
1211:
1208:
1205:
1200:
1196:
1192:
1188:
1184:
1181:
1145:
1142:
1139:
1128:
1127:
1115:
1110:
1107:
1104:
1100:
1096:
1092:
1089:
1086:
1083:
1080:
1077:
1054:
1031:Big-O notation
1018:
1007:
1006:
995:
991:
986:
981:
977:
974:
971:
968:
965:
962:
935:
924:
923:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
831:
828:
805:
804:
792:
787:
782:
778:
775:
753:
749:
745:
741:
736:
733:
728:
722:
717:
714:
711:
707:
703:
700:
696:
691:
688:
683:
677:
672:
669:
666:
662:
658:
655:
652:
649:
646:
623:
618:
613:
610:
574:
537:
531:
530:
519:
516:
513:
508:
505:
502:
498:
492:
489:
486:
483:
479:
473:
470:
467:
463:
459:
456:
453:
450:
445:
441:
435:
432:
429:
425:
421:
418:
415:
412:
407:
403:
380:
379:
368:
361:
358:
355:
352:
347:
344:
341:
335:
331:
328:
325:
322:
317:
313:
309:
306:
303:
300:
297:
283:
282:
271:
264:
261:
258:
255:
250:
247:
244:
238:
234:
231:
228:
225:
222:
217:
214:
211:
207:
203:
200:
197:
194:
191:
176:
173:
134:
130:
126:
123:
94:
90:
86:
83:
54:
50:
46:
43:
26:
9:
6:
4:
3:
2:
4160:
4149:
4146:
4144:
4141:
4139:
4136:
4135:
4133:
4123:
4119:
4116:
4113:
4109:
4107:
4103:
4099:
4095:
4091:
4087:
4081:
4077:
4073:
4069:
4067:
4063:
4059:
4057:
4056:0-486-41740-9
4053:
4049:
4045:
4042:
4041:
4029:
4024:
4015:
4007:
4003:
3999:
3997:0-8218-0737-4
3993:
3989:
3985:
3981:
3975:
3968:
3964:
3960:
3956:
3952:
3947:
3942:
3938:
3934:
3930:
3926:
3920:
3912:
3908:
3904:
3900:
3896:
3892:
3888:
3884:
3880:
3879:Huxley, M. N.
3874:
3872:
3862:
3857:
3853:
3849:
3845:
3841:
3837:
3831:
3823:
3821:0-486-42813-3
3817:
3813:
3806:
3804:
3802:
3800:
3798:
3796:
3794:
3785:
3779:
3775:
3771:
3765:
3757:
3751:
3747:
3743:
3742:R. C. Vaughan
3739:
3733:
3731:
3726:
3718:
3704:
3701:
3698:
3675:
3667:
3639:
3636:
3630:
3625:
3621:
3612:
3604:
3600:
3591:
3588:
3585:
3577:
3574:
3571:
3567:
3560:
3557:
3554:
3550:
3545:
3539:
3531:
3527:
3519:
3518:
3517:
3515:
3511:
3495:
3492:
3489:
3466:
3460:
3440:
3437:
3428:
3424:
3421:
3398:
3395:
3389:
3384:
3380:
3371:
3363:
3359:
3350:
3347:
3338:
3329:
3326:
3317:
3312:
3305:
3302:
3299:
3295:
3290:
3284:
3271:
3270:
3269:
3267:
3248:
3242:
3222:
3219:
3216:
3193:
3190:
3184:
3179:
3175:
3166:
3158:
3154:
3145:
3142:
3139:
3131:
3128:
3125:
3121:
3114:
3111:
3108:
3104:
3099:
3093:
3087:
3080:
3079:
3078:
3076:
3052:
3043:
3040:
3035:
3032:
3029:
3023:
3018:
3014:
3005:
3002:
3001:
2979:
2976:
2973:
2963:
2960:
2955:
2952:
2949:
2946:
2940:
2938:
2931:
2927:
2916:
2913:
2910:
2907:
2904:
2894:
2891:
2886:
2883:
2880:
2877:
2871:
2869:
2862:
2858:
2847:
2844:
2841:
2838:
2835:
2825:
2822:
2819:
2814:
2811:
2808:
2802:
2800:
2793:
2789:
2778:
2775:
2772:
2769:
2766:
2756:
2753:
2750:
2745:
2742:
2739:
2733:
2731:
2724:
2720:
2710:
2707:
2702:
2697:
2693:
2688:
2680:
2677:
2672:
2667:
2663:
2658:
2650:
2647:
2642:
2640:
2633:
2629:
2618:
2615:
2612:
2609:
2606:
2596:
2593:
2588:
2585:
2582:
2579:
2573:
2571:
2564:
2560:
2548:
2547:
2546:
2528:
2520:
2517:
2512:
2507:
2503:
2495:
2494:
2493:
2476:
2468:
2465:
2460:
2455:
2451:
2443:
2442:
2441:
2427:
2405:
2401:
2380:
2377:
2374:
2353:
2348:
2345:
2340:
2336:
2331:
2327:
2323:
2320:
2314:
2306:
2279:
2275:
2266:
2250:
2230:
2227:
2224:
2204:
2201:
2198:
2174:
2170:
2167:
2162:
2159:
2156:
2152:
2146:
2142:
2138:
2135:
2132:
2128:
2123:
2119:
2116:
2110:
2102:
2090:
2089:
2088:
2074:
2071:
2068:
2061:
2043:
2039:
2011:
2003:
1995:
1989:
1986:
1983:
1975:
1971:
1967:
1964:
1958:
1950:
1946:
1938:
1937:
1936:
1928:
1926:
1908:
1904:
1900:
1896:
1891:
1884:
1858:
1831:
1828:
1825:
1821:
1817:
1814:
1811:
1800:
1796:
1776:
1771:
1768:
1765:
1761:
1757:
1754:
1751:
1740:
1736:
1732:
1712:
1707:
1704:
1701:
1697:
1693:
1690:
1687:
1676:
1661:
1658:
1655:
1651:
1647:
1644:
1641:
1630:
1610:
1605:
1602:
1599:
1595:
1591:
1588:
1585:
1574:
1570:
1555:
1552:
1549:
1545:
1541:
1538:
1535:
1524:
1523:H. E. Richert
1520:
1516:
1496:
1491:
1488:
1485:
1481:
1477:
1474:
1471:
1460:
1445:
1442:
1439:
1435:
1431:
1428:
1425:
1414:
1410:
1393:
1389:
1385:
1381:
1377:
1374:
1371:
1365:
1351:
1333:
1329:
1325:
1321:
1317:
1314:
1308:
1294:
1278:
1258:
1254:
1250:
1247:
1244:
1233:
1229:
1215:
1209:
1206:
1203:
1198:
1194:
1190:
1186:
1179:
1171:
1167:
1166:
1165:
1163:
1159:
1143:
1140:
1137:
1113:
1108:
1105:
1102:
1098:
1094:
1090:
1087:
1081:
1068:
1067:
1066:
1052:
1044:
1041:in 1849. The
1040:
1036:
1032:
1016:
993:
989:
984:
979:
975:
972:
966:
953:
952:
951:
949:
933:
904:
895:
889:
886:
883:
880:
874:
871:
868:
865:
862:
859:
856:
850:
844:
837:
836:
835:
827:
823:
818:
812:
810:
790:
785:
780:
776:
773:
751:
747:
743:
739:
734:
731:
726:
720:
715:
712:
709:
705:
701:
698:
694:
689:
686:
681:
675:
670:
667:
664:
660:
656:
650:
644:
637:
636:
635:
616:
608:
600:
596:
592:
588:
585: =
584:
580:
573:
569:
565:
561:
557:
553:
549:
545:
540:
536:
514:
506:
503:
500:
496:
490:
487:
484:
481:
477:
471:
468:
465:
461:
457:
451:
443:
439:
433:
430:
427:
423:
419:
413:
405:
401:
393:
392:
391:
389:
385:
366:
359:
356:
353:
350:
345:
342:
339:
333:
329:
323:
315:
311:
307:
301:
295:
288:
287:
286:
269:
262:
259:
256:
253:
248:
245:
242:
236:
232:
226:
220:
215:
212:
209:
205:
201:
195:
189:
182:
181:
180:
172:
170:
166:
162:
158:
154:
153:number theory
132:
128:
124:
121:
112:
92:
88:
84:
81:
72:
52:
48:
44:
41:
32:
19:
4121:
4111:
4110:H. E. Rose.
4105:
4075:
4065:
4061:
4047:
4044:H.M. Edwards
4023:
4014:
3983:
3974:
3966:
3936:
3932:
3919:
3886:
3882:
3851:
3847:
3830:
3811:
3773:
3764:
3745:
3654:
3413:
3208:
3072:
2544:
2491:
2217:. As in the
2191:for integer
2190:
2030:
1934:
1847:
1741:proved that
1525:proved that
1349:
1292:
1234:showed that
1161:
1129:
1042:
1008:
925:
833:
813:
806:
598:
594:
586:
582:
578:
571:
567:
563:
559:
555:
551:
547:
543:
538:
534:
532:
387:
381:
284:
178:
168:
156:
150:
4118:M.N. Huxley
3836:Iwaniec, H.
2265:Adolf Piltz
1799:M.N. Huxley
1232:G. H. Hardy
4132:Categories
4102:0335.10001
4038:References
4006:0814.11001
3911:1065.11079
1735:H. Iwaniec
1352:for which
1295:for which
1170:G. Voronoi
1065:for which
175:Definition
3955:0065-1036
3903:0024-6115
3854:: 60–93.
3702:≥
3664:Δ
3601:ζ
3595:∞
3581:∞
3575:−
3568:∫
3558:π
3433:′
3360:ζ
3354:∞
3343:′
3333:∞
3327:−
3322:′
3313:∫
3303:π
3279:Δ
3243:ζ
3155:ζ
3149:∞
3135:∞
3129:−
3122:∫
3112:π
3033:−
3015:α
2977:≥
2953:−
2941:≤
2928:α
2914:≤
2908:≤
2884:−
2872:≤
2859:α
2845:≤
2839:≤
2812:−
2803:≤
2790:α
2776:≤
2770:≤
2743:−
2734:≤
2721:α
2703:≤
2694:α
2673:≤
2664:α
2643:≤
2630:α
2616:≤
2610:≤
2586:−
2574:≤
2561:α
2513:≤
2504:α
2461:≤
2452:α
2428:θ
2402:α
2375:ε
2349:ε
2337:α
2303:Δ
2276:α
2202:≥
2168:
2160:−
2136:−
2099:Δ
2072:−
2000:Δ
1987:
1879:Δ
1859:θ
1815:≤
1812:θ
1797:In 2003,
1780:¯
1755:≤
1752:θ
1733:In 1988,
1716:¯
1691:≤
1688:θ
1645:≤
1642:θ
1614:¯
1589:≤
1586:θ
1571:In 1969,
1539:≤
1536:θ
1517:In 1950,
1500:¯
1475:≤
1472:θ
1429:≤
1426:θ
1411:In 1922,
1375:−
1360:Δ
1303:Δ
1248:≥
1245:θ
1230:In 1916,
1207:
1168:In 1904,
1138:ϵ
1109:ϵ
1103:θ
1076:Δ
1053:θ
1039:Dirichlet
961:Δ
934:γ
899:Δ
887:−
884:γ
866:
744:−
706:∑
661:∑
504:−
488:≤
478:∑
469:≤
462:∑
431:≤
424:∑
334:∑
312:σ
260:≤
237:∑
213:≤
206:∑
4074:(1976),
3982:(1994).
3963:59450869
3927:(1992).
3842:(1988).
3772:(2004).
3744:(2007).
3235:. Here,
1029:denotes
791:⌋
781:⌊
766:, where
740:⌋
727:⌊
695:⌋
682:⌊
4094:0434929
3510:residue
3264:is the
2420:is the
946:is the
820:in the
817:A006218
591:simplex
382:is the
4100:
4092:
4082:
4054:
4004:
3994:
3961:
3953:
3909:
3901:
3818:
3780:
3752:
3691:, for
3050:
2686:
2656:
2526:
2474:
2031:where
1009:Here,
926:where
911:
634:time:
533:where
285:where
155:, the
3959:S2CID
3721:Notes
3512:, by
3414:with
2058:is a
1497:29268
4080:ISBN
4052:ISBN
3992:ISBN
3951:ISSN
3899:ISSN
3816:ISBN
3778:ISBN
3750:ISBN
3438:<
3425:<
3220:>
3209:for
2378:>
1848:So,
1737:and
1708:0.32
1656:1067
1446:0.33
1372:<
1315:>
1141:>
822:OEIS
570:) =
562:=2,
125:<
85:<
45:<
4098:Zbl
4002:Zbl
3941:doi
3907:Zbl
3891:doi
3856:doi
2541:and
2469:416
2466:131
2153:log
1984:log
1856:inf
1826:416
1818:131
1809:inf
1772:0.3
1749:inf
1713:407
1702:108
1685:inf
1648:346
1639:inf
1611:324
1583:inf
1533:inf
1492:0.3
1469:inf
1440:100
1423:inf
1242:inf
1204:log
863:log
151:In
4134::
4096:,
4090:MR
4088:,
4046:,
4000:.
3965:.
3957:.
3949:.
3937:60
3935:.
3931:.
3905:.
3897:.
3887:87
3885:.
3870:^
3852:29
3850:.
3846:.
3838:;
3792:^
3740:;
3729:^
3717:.
3077::
2980:58
2956:34
2917:57
2905:51
2892:32
2887:98
2878:31
2848:50
2836:26
2779:25
2767:12
2711:10
2698:11
2681:60
2678:41
2668:10
2651:54
2648:35
2521:96
2518:43
1927:.
1777:18
1766:22
1694:35
1606:0.
1600:37
1592:12
1550:46
1542:15
1486:82
1478:27
1432:33
824:):
811:.
583:jk
171:.
129:10
89:10
49:10
4008:.
3943::
3913:.
3893::
3864:.
3858::
3824:.
3786:.
3758:.
3705:2
3699:k
3679:)
3676:x
3673:(
3668:k
3640:w
3637:d
3631:w
3626:w
3622:x
3616:)
3613:w
3610:(
3605:k
3592:i
3589:+
3586:c
3578:i
3572:c
3561:i
3555:2
3551:1
3546:=
3543:)
3540:x
3537:(
3532:k
3528:D
3496:1
3493:=
3490:w
3470:)
3467:x
3464:(
3461:D
3441:1
3429:c
3422:0
3399:w
3396:d
3390:w
3385:w
3381:x
3375:)
3372:w
3369:(
3364:2
3351:i
3348:+
3339:c
3330:i
3318:c
3306:i
3300:2
3296:1
3291:=
3288:)
3285:x
3282:(
3252:)
3249:s
3246:(
3223:1
3217:c
3194:w
3191:d
3185:w
3180:w
3176:x
3170:)
3167:w
3164:(
3159:2
3146:i
3143:+
3140:c
3132:i
3126:c
3115:i
3109:2
3105:1
3100:=
3097:)
3094:x
3091:(
3088:D
3053:.
3044:k
3041:2
3036:1
3030:k
3024:=
3019:k
2983:)
2974:k
2971:(
2964:k
2961:7
2950:k
2947:7
2932:k
2920:)
2911:k
2902:(
2895:k
2881:k
2863:k
2851:)
2842:k
2833:(
2826:4
2823:+
2820:k
2815:1
2809:k
2794:k
2782:)
2773:k
2764:(
2757:2
2754:+
2751:k
2746:2
2740:k
2725:k
2708:7
2689:,
2659:,
2634:9
2622:)
2619:8
2613:k
2607:4
2604:(
2597:k
2594:4
2589:4
2583:k
2580:3
2565:k
2529:,
2508:3
2477:,
2456:2
2406:2
2381:0
2354:)
2346:+
2341:k
2332:x
2328:(
2324:O
2321:=
2318:)
2315:x
2312:(
2307:k
2280:k
2251:k
2231:2
2228:=
2225:k
2205:2
2199:k
2175:)
2171:x
2163:2
2157:k
2147:k
2143:/
2139:1
2133:1
2129:x
2124:(
2120:O
2117:=
2114:)
2111:x
2108:(
2103:k
2075:1
2069:k
2044:k
2040:P
2015:)
2012:x
2009:(
2004:k
1996:+
1993:)
1990:x
1981:(
1976:k
1972:P
1968:x
1965:=
1962:)
1959:x
1956:(
1951:k
1947:D
1909:4
1905:/
1901:1
1897:x
1892:/
1888:)
1885:x
1882:(
1844:.
1829:=
1822:/
1794:.
1769:=
1762:/
1758:7
1730:.
1705:=
1698:/
1674:.
1659:=
1652:/
1628:.
1603:=
1596:/
1568:.
1553:=
1546:/
1514:.
1489:=
1482:/
1458:.
1443:=
1436:/
1408:.
1394:4
1390:/
1386:1
1382:x
1378:K
1369:)
1366:x
1363:(
1350:x
1334:4
1330:/
1326:1
1322:x
1318:K
1312:)
1309:x
1306:(
1293:x
1279:K
1259:4
1255:/
1251:1
1216:.
1213:)
1210:x
1199:3
1195:/
1191:1
1187:x
1183:(
1180:O
1144:0
1114:)
1106:+
1099:x
1095:(
1091:O
1088:=
1085:)
1082:x
1079:(
1017:O
994:.
990:)
985:x
980:(
976:O
973:=
970:)
967:x
964:(
908:)
905:x
902:(
896:+
893:)
890:1
881:2
878:(
875:x
872:+
869:x
860:x
857:=
854:)
851:x
848:(
845:D
786:x
777:=
774:u
752:2
748:u
735:k
732:x
721:u
716:1
713:=
710:k
702:2
699:=
690:k
687:x
676:x
671:1
668:=
665:k
657:=
654:)
651:x
648:(
645:D
622:)
617:x
612:(
609:O
599:x
597:(
595:D
587:x
579:x
577:(
575:2
572:D
568:x
566:(
564:D
560:k
556:k
552:k
548:n
544:n
542:(
539:k
535:d
518:)
515:n
512:(
507:1
501:k
497:d
491:x
485:n
482:m
472:x
466:m
458:=
455:)
452:n
449:(
444:k
440:d
434:x
428:n
420:=
417:)
414:x
411:(
406:k
402:D
388:n
367:1
360:n
357:=
354:k
351:j
346:k
343:,
340:j
330:=
327:)
324:n
321:(
316:0
308:=
305:)
302:n
299:(
296:d
270:1
263:x
257:k
254:j
249:k
246:,
243:j
233:=
230:)
227:n
224:(
221:d
216:x
210:n
202:=
199:)
196:x
193:(
190:D
133:7
122:x
93:7
82:x
53:4
42:x
20:)
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