31:
38:, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle.
1626:
1965:
2036:
is physical device for estimating the area of shapes based on the same principle. It consists of a square grid of dots, printed on a transparent sheet; the area of a shape can be estimated as the product of the number of dots in the shape with the area of a grid square.
2572:
1481:
927:
1425:
2367:
1206:
1795:
1296:
2142:, as it involves searching for primitive solutions to the original circle problem. It can be intuitively understood as the question of how many trees within a distance of r are visible in the
1784:
1087:
2133:
350:
695:
2404:
2628:
2654:
283:
2771:(2002). "Integer points, exponential sums and the Riemann zeta function". In Bennett, M. A.; Berndt, B. C.; Boston, N.; Diamond, H. G.; Hildebrand, A. J.; Philipp, W. (eds.).
124:
1344:
762:
2372:
As with the usual circle problem, the problematic part of the primitive circle problem is reducing the exponent in the error term. At present, the best known exponent is
2281:
631:
577:
483:
1679:
994:
603:. This is because on average, each unit square contains one lattice point. Thus, the actual number of lattice points in the circle is approximately equal to its area,
817:
150:
1020:
2416:
1995:
182:
2202:
2173:
1469:
956:
727:
547:
431:
402:
2073:
2595:
2223:
1699:
1364:
782:
601:
452:
373:
222:
202:
79:
1621:{\displaystyle N(r)=1+4\sum _{i=0}^{\infty }\left(\left\lfloor {\frac {r^{2}}{4i+1}}\right\rfloor -\left\lfloor {\frac {r^{2}}{4i+3}}\right\rfloor \right).}
2025:
or other objects. There is an extensive literature on these problems. If one ignores the geometry and merely considers the problem an algebraic one of
822:
3023:
2237:
509:
496:
2773:
Number theory for the millennium, II: Papers from the conference held at the
University of Illinois at Urbana–Champaign, Urbana, IL, May 21–26, 2000
1372:
2289:
2009:
Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapes, for example
2892:
1960:{\displaystyle N(x)-{\frac {r_{2}(x^{2})}{2}}=\pi x^{2}+x\sum _{n=1}^{\infty }{\frac {r_{2}(n)}{\sqrt {n}}}J_{1}(2\pi x{\sqrt {n}}),}
1105:
2828:
2744:
1221:
2014:
1632:
3013:
2924:
2021:. Similarly one could extend the question from two dimensions to higher dimensions, and ask for integer points within a
2802:
1707:
229:
1031:
2026:
2081:
3018:
1643:
299:
639:
2837:
2375:
2731:. Problem Books in Mathematics. Vol. 1 (3rd ed.). New York: Springer-Verlag. pp. 365–367.
2029:, then there one could increase the exponents appearing in the problem from squares to cubes, or higher.
81:. This number is approximated by the area of the circle, so the real problem is to accurately bound the
2600:
1636:
85:
describing how the number of points differs from the area. The first progress on a solution was made by
2633:
235:
100:
1304:
86:
732:
2842:
2143:
30:
2949:
2855:
2826:
Hirschhorn, Michael D. (2000). "Partial fractions and four classical theorems of number theory".
2798:
2407:
2251:
1789:
Most recent progress rests on the following
Identity, which has been first discovered by Hardy:
606:
552:
458:
2972:
2740:
2567:{\displaystyle V(r)={\frac {6}{\pi }}r^{2}+O(r\exp(-c(\log r)^{3/5}(\log \log r^{2})^{-1/5}))}
1648:
961:
787:
129:
3028:
2933:
2847:
2732:
2245:
999:
2945:
2812:
2780:
2754:
2688:
1973:
155:
17:
2941:
2808:
2776:
2750:
2684:
2178:
2149:
1998:
1445:
932:
703:
523:
407:
378:
51:
2975:
2052:
152:. Gauss's circle problem asks how many points there are inside this circle of the form
2888:
2724:
2580:
2208:
2033:
1684:
1472:
1349:
1212:
767:
586:
580:
437:
358:
207:
187:
64:
3007:
2953:
2794:
2768:
1431:
1096:
2410:. Without assuming the Riemann hypothesis, the best upper bound currently known is
1430:
with the lower bound from Hardy and Landau in 1915, and the upper bound proved by
922:{\displaystyle N({\sqrt {17}})=57,N({\sqrt {18}})=61,N({\sqrt {20}})=69,N(5)=81.}
2996:
2702:
Hardy, G. H. (1915). "On the expression of a number as the sum of two squares".
2676:
1092:
43:
2991:
2736:
82:
2244:
Using the same ideas as the usual Gauss circle problem and the fact that the
2980:
2018:
2937:
2017:
is the equivalent problem where the circle is replaced by the rectangular
2022:
1420:{\displaystyle {\frac {1}{2}}<t\leq {\frac {131}{208}}=0.6298\ldots ,}
225:
2859:
2046:
729:
of relatively small absolute value. Finding a correct upper bound for
2681:
Ramanujan: Twelve
Lectures on Subjects Suggested by His Life and Work
2146:, standing in the origin. If the number of such solutions is denoted
59:
55:
2851:
2683:(3rd ed.). New York: Chelsea Publishing Company. p. 67.
2362:{\displaystyle V(r)={\frac {6}{\pi }}r^{2}+O(r^{1+\varepsilon }).}
2010:
502:
0, 3, 13, 28, 50, 79, 113, 154, 201, 254, 314, 380, 452 (sequence
489:
1, 5, 13, 29, 49, 81, 113, 149, 197, 253, 317, 377, 441 (sequence
29:
1471:
can be given by several series. In terms of a sum involving the
285:, the question is equivalently asking how many pairs of integers
2656:
is currently known that does not assume the
Riemann Hypothesis.
2807:. New York, N. Y.: Chelsea Publishing Company. pp. 37–38.
1201:{\displaystyle |E(r)|\neq o\left(r^{1/2}(\log r)^{1/4}\right),}
2232:
504:
491:
455:
an integer between 0 and 12 followed by the list of values
1291:{\displaystyle |E(r)|=O\left(r^{1/2+\varepsilon }\right).}
2597:. In particular, no bound on the error term of the form
2775:. Natick, Massachusetts: A K Peters. pp. 275–290.
2230:
0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192 … (sequence
1681:
is defined as the number of ways of writing the number
34:
A circle of radius 5 centered at the origin has area 25
404:
then the following list shows the first few values of
2636:
2603:
2583:
2419:
2378:
2292:
2254:
2211:
2181:
2152:
2084:
2055:
2045:
Another generalization is to calculate the number of
1976:
1798:
1710:
1687:
1651:
1484:
1448:
1375:
1352:
1307:
1224:
1108:
1034:
1002:
964:
935:
825:
790:
770:
735:
706:
642:
609:
589:
555:
526:
461:
440:
410:
381:
361:
302:
238:
210:
190:
158:
132:
103:
67:
27:
How many integer lattice points there are in a circle
2922:Wu, Jie (2002). "On the primitive circle problem".
2648:
2622:
2589:
2566:
2398:
2361:
2275:
2217:
2196:
2167:
2127:
2067:
1989:
1959:
1778:
1693:
1673:
1620:
1463:
1419:
1358:
1338:
1290:
1200:
1081:
1014:
988:
950:
921:
811:
776:
764:is thus the form the problem has taken. Note that
756:
721:
689:
625:
595:
571:
541:
477:
446:
425:
396:
367:
344:
277:
216:
196:
176:
144:
118:
73:
1779:{\displaystyle N(r)=\sum _{n=0}^{r^{2}}r_{2}(n).}
2283:, it is relatively straightforward to show that
1215:. It is conjectured that the correct bound is
2877:. Vol. 2. Verlag S. Hirzel. p. 189.
1082:{\displaystyle |E(r)|\leq 2{\sqrt {2}}\pi r.}
8:
2727:(2004). "F1: Gauß's lattice point problem".
1635:, which follows almost immediately from the
2841:
2635:
2608:
2602:
2582:
2548:
2541:
2531:
2502:
2498:
2449:
2435:
2418:
2382:
2377:
2341:
2322:
2308:
2291:
2267:
2258:
2253:
2246:probability that two integers are coprime
2210:
2180:
2151:
2124:
2115:
2102:
2089:
2083:
2054:
1981:
1975:
1944:
1926:
1899:
1892:
1886:
1875:
1859:
1834:
1821:
1814:
1797:
1758:
1746:
1741:
1730:
1709:
1686:
1656:
1650:
1584:
1578:
1545:
1539:
1524:
1513:
1483:
1447:
1395:
1376:
1374:
1351:
1330:
1306:
1265:
1261:
1242:
1225:
1223:
1180:
1176:
1150:
1146:
1126:
1109:
1107:
1063:
1052:
1035:
1033:
1001:
963:
934:
882:
857:
832:
824:
789:
769:
734:
705:
686:
665:
641:
617:
608:
588:
563:
554:
525:
469:
460:
439:
409:
380:
360:
333:
320:
307:
301:
269:
256:
243:
237:
209:
189:
157:
131:
110:
106:
105:
102:
66:
2128:{\displaystyle m^{2}+n^{2}\leq r^{2}.\,}
2671:
2669:
2665:
996:after which it decreases (at a rate of
784:does not have to be an integer. After
50:is the problem of determining how many
2917:
2915:
2913:
345:{\displaystyle m^{2}+n^{2}\leq r^{2}.}
2719:
2717:
690:{\displaystyle N(r)=\pi r^{2}+E(r)\,}
126:with center at the origin and radius
7:
2399:{\displaystyle 221/304+\varepsilon }
1099:found a lower bound by showing that
1022:) until the next time it increases.
517:Bounds on a solution and conjecture
2729:Unsolved Problems in Number Theory
2623:{\displaystyle r^{1-\varepsilon }}
1887:
1642:A much simpler sum appears if the
1525:
25:
2992:"Pi hiding in prime regularities"
2829:The American Mathematical Monthly
2649:{\displaystyle \varepsilon >0}
2226:taking small integer values are
633:. So it should be expected that
278:{\displaystyle x^{2}+y^{2}=r^{2}}
3024:Unsolved problems in mathematics
2704:Quarterly Journal of Mathematics
2001:of the first kind with order 1.
1701:as the sum of two squares. Then
485:rounded to the nearest integer:
119:{\displaystyle \mathbb {R} ^{2}}
58:centered at the origin and with
1339:{\displaystyle E(r)\leq Cr^{t}}
2900:Przegląd Matematyczno-Fizyczny
2875:Vorlesungen über Zahlentheorie
2561:
2558:
2538:
2512:
2495:
2482:
2473:
2461:
2429:
2423:
2353:
2334:
2302:
2296:
2191:
2185:
2162:
2156:
1951:
1932:
1911:
1905:
1840:
1827:
1808:
1802:
1770:
1764:
1720:
1714:
1668:
1662:
1494:
1488:
1458:
1452:
1317:
1311:
1243:
1239:
1233:
1226:
1173:
1160:
1127:
1123:
1117:
1110:
1053:
1049:
1043:
1036:
945:
939:
910:
904:
889:
879:
864:
854:
839:
829:
800:
794:
757:{\displaystyle \mid E(r)\mid }
751:
748:
742:
736:
716:
710:
683:
677:
652:
646:
536:
530:
420:
414:
391:
385:
224:are both integers. Since the
171:
159:
1:
2138:This problem is known as the
2804:Geometry and the Imagination
2041:The primitive circle problem
1025:Gauss managed to prove that
2015:Dirichlet's divisor problem
1633:Jacobi's two-square theorem
228:of this circle is given in
18:Gauss's circle problem
3045:
2925:Monatshefte für Mathematik
2893:"O mierzeniu pól płaskich"
2276:{\displaystyle 6/\pi ^{2}}
355:If the answer for a given
2737:10.1007/978-0-387-26677-0
1631:This is a consequence of
626:{\displaystyle \pi r^{2}}
572:{\displaystyle \pi r^{2}}
478:{\displaystyle \pi r^{2}}
2976:"Gauss's circle problem"
2577:for a positive constant
2140:primitive circle problem
2027:Diophantine inequalities
1674:{\displaystyle r_{2}(n)}
1475:it can be expressed as:
1346:, the current bounds on
989:{\displaystyle 8,4,8,12}
2873:Landau, Edmund (1927).
1644:sum of squares function
812:{\displaystyle N(4)=49}
145:{\displaystyle r\geq 0}
2650:
2624:
2591:
2568:
2400:
2363:
2277:
2219:
2198:
2169:
2129:
2069:
1991:
1961:
1891:
1780:
1753:
1695:
1675:
1622:
1529:
1465:
1421:
1360:
1340:
1292:
1202:
1083:
1016:
1015:{\displaystyle 2\pi r}
990:
952:
923:
813:
778:
758:
723:
691:
627:
597:
573:
543:
479:
448:
427:
398:
369:
346:
279:
218:
198:
178:
146:
120:
75:
54:points there are in a
39:
2938:10.1007/s006050200006
2651:
2625:
2592:
2569:
2401:
2364:
2278:
2220:
2199:
2170:
2130:
2070:
1992:
1990:{\displaystyle J_{1}}
1962:
1871:
1781:
1726:
1696:
1676:
1637:Jacobi triple product
1623:
1509:
1466:
1422:
1361:
1341:
1293:
1203:
1084:
1017:
991:
953:
924:
814:
779:
759:
724:
692:
628:
598:
574:
544:
480:
449:
428:
399:
370:
347:
280:
230:Cartesian coordinates
219:
199:
179:
177:{\displaystyle (m,n)}
147:
121:
97:Consider a circle in
76:
33:
3014:Arithmetic functions
2634:
2601:
2581:
2417:
2376:
2290:
2252:
2209:
2197:{\displaystyle V(r)}
2179:
2168:{\displaystyle V(r)}
2150:
2082:
2053:
1974:
1796:
1708:
1685:
1649:
1482:
1464:{\displaystyle N(r)}
1446:
1373:
1350:
1305:
1222:
1106:
1095:and, independently,
1032:
1000:
962:
951:{\displaystyle E(r)}
933:
823:
788:
768:
733:
722:{\displaystyle E(r)}
704:
700:for some error term
640:
607:
587:
581:area inside a circle
553:
542:{\displaystyle N(r)}
524:
459:
438:
426:{\displaystyle N(r)}
408:
397:{\displaystyle N(r)}
379:
359:
300:
293:there are such that
236:
208:
188:
156:
130:
101:
87:Carl Friedrich Gauss
65:
48:Gauss circle problem
2406:if one assumes the
2175:then the values of
2068:{\displaystyle m,n}
2973:Weisstein, Eric W.
2646:
2620:
2587:
2564:
2408:Riemann hypothesis
2396:
2359:
2273:
2215:
2194:
2165:
2125:
2075:to the inequality
2065:
2049:integer solutions
1987:
1957:
1776:
1691:
1671:
1618:
1461:
1417:
1356:
1336:
1288:
1198:
1079:
1012:
986:
948:
919:
809:
774:
754:
719:
687:
623:
593:
569:
539:
475:
444:
423:
394:
365:
342:
275:
214:
194:
174:
142:
116:
89:, hence its name.
71:
40:
2990:Grant Sanderson,
2590:{\displaystyle c}
2443:
2316:
2218:{\displaystyle r}
1949:
1920:
1919:
1847:
1694:{\displaystyle n}
1604:
1565:
1403:
1384:
1359:{\displaystyle t}
1213:little o-notation
1068:
887:
862:
837:
777:{\displaystyle r}
596:{\displaystyle r}
447:{\displaystyle r}
368:{\displaystyle r}
217:{\displaystyle n}
197:{\displaystyle m}
74:{\displaystyle r}
16:(Redirected from
3036:
3000:
2986:
2985:
2958:
2957:
2919:
2908:
2907:
2897:
2885:
2879:
2878:
2870:
2864:
2863:
2845:
2823:
2817:
2816:
2791:
2785:
2784:
2765:
2759:
2758:
2721:
2712:
2711:
2699:
2693:
2692:
2673:
2655:
2653:
2652:
2647:
2629:
2627:
2626:
2621:
2619:
2618:
2596:
2594:
2593:
2588:
2573:
2571:
2570:
2565:
2557:
2556:
2552:
2536:
2535:
2511:
2510:
2506:
2454:
2453:
2444:
2436:
2405:
2403:
2402:
2397:
2386:
2368:
2366:
2365:
2360:
2352:
2351:
2327:
2326:
2317:
2309:
2282:
2280:
2279:
2274:
2272:
2271:
2262:
2235:
2224:
2222:
2221:
2216:
2203:
2201:
2200:
2195:
2174:
2172:
2171:
2166:
2144:Euclid's orchard
2134:
2132:
2131:
2126:
2120:
2119:
2107:
2106:
2094:
2093:
2074:
2072:
2071:
2066:
1996:
1994:
1993:
1988:
1986:
1985:
1966:
1964:
1963:
1958:
1950:
1945:
1931:
1930:
1921:
1915:
1914:
1904:
1903:
1893:
1890:
1885:
1864:
1863:
1848:
1843:
1839:
1838:
1826:
1825:
1815:
1785:
1783:
1782:
1777:
1763:
1762:
1752:
1751:
1750:
1740:
1700:
1698:
1697:
1692:
1680:
1678:
1677:
1672:
1661:
1660:
1627:
1625:
1624:
1619:
1614:
1610:
1609:
1605:
1603:
1589:
1588:
1579:
1570:
1566:
1564:
1550:
1549:
1540:
1528:
1523:
1470:
1468:
1467:
1462:
1426:
1424:
1423:
1418:
1404:
1396:
1385:
1377:
1365:
1363:
1362:
1357:
1345:
1343:
1342:
1337:
1335:
1334:
1297:
1295:
1294:
1289:
1284:
1280:
1279:
1269:
1246:
1229:
1207:
1205:
1204:
1199:
1194:
1190:
1189:
1188:
1184:
1159:
1158:
1154:
1130:
1113:
1088:
1086:
1085:
1080:
1069:
1064:
1056:
1039:
1021:
1019:
1018:
1013:
995:
993:
992:
987:
957:
955:
954:
949:
929:At these places
928:
926:
925:
920:
888:
883:
863:
858:
838:
833:
818:
816:
815:
810:
783:
781:
780:
775:
763:
761:
760:
755:
728:
726:
725:
720:
696:
694:
693:
688:
670:
669:
632:
630:
629:
624:
622:
621:
602:
600:
599:
594:
578:
576:
575:
570:
568:
567:
548:
546:
545:
540:
507:
494:
484:
482:
481:
476:
474:
473:
453:
451:
450:
445:
432:
430:
429:
424:
403:
401:
400:
395:
374:
372:
371:
366:
351:
349:
348:
343:
338:
337:
325:
324:
312:
311:
284:
282:
281:
276:
274:
273:
261:
260:
248:
247:
223:
221:
220:
215:
203:
201:
200:
195:
183:
181:
180:
175:
151:
149:
148:
143:
125:
123:
122:
117:
115:
114:
109:
80:
78:
77:
72:
37:
21:
3044:
3043:
3039:
3038:
3037:
3035:
3034:
3033:
3004:
3003:
2989:
2971:
2970:
2967:
2962:
2961:
2921:
2920:
2911:
2895:
2889:Steinhaus, Hugo
2887:
2886:
2882:
2872:
2871:
2867:
2852:10.2307/2589321
2825:
2824:
2820:
2799:Cohn-Vossen, S.
2793:
2792:
2788:
2767:
2766:
2762:
2747:
2725:Guy, Richard K.
2723:
2722:
2715:
2701:
2700:
2696:
2675:
2674:
2667:
2662:
2632:
2631:
2604:
2599:
2598:
2579:
2578:
2537:
2527:
2494:
2445:
2415:
2414:
2374:
2373:
2337:
2318:
2288:
2287:
2263:
2250:
2249:
2231:
2207:
2206:
2177:
2176:
2148:
2147:
2111:
2098:
2085:
2080:
2079:
2051:
2050:
2043:
2007:
2005:Generalizations
1999:Bessel function
1977:
1972:
1971:
1922:
1895:
1894:
1855:
1830:
1817:
1816:
1794:
1793:
1754:
1742:
1706:
1705:
1683:
1682:
1652:
1647:
1646:
1590:
1580:
1574:
1551:
1541:
1535:
1534:
1530:
1480:
1479:
1444:
1443:
1440:
1371:
1370:
1348:
1347:
1326:
1303:
1302:
1257:
1253:
1220:
1219:
1172:
1142:
1141:
1137:
1104:
1103:
1030:
1029:
998:
997:
960:
959:
931:
930:
821:
820:
786:
785:
766:
765:
731:
730:
702:
701:
661:
638:
637:
613:
605:
604:
585:
584:
559:
551:
550:
522:
521:
519:
503:
490:
465:
457:
456:
436:
435:
406:
405:
377:
376:
357:
356:
329:
316:
303:
298:
297:
265:
252:
239:
234:
233:
206:
205:
186:
185:
154:
153:
128:
127:
104:
99:
98:
95:
63:
62:
52:integer lattice
35:
28:
23:
22:
15:
12:
11:
5:
3042:
3040:
3032:
3031:
3026:
3021:
3019:Lattice points
3016:
3006:
3005:
3002:
3001:
2987:
2966:
2965:External links
2963:
2960:
2959:
2909:
2880:
2865:
2843:10.1.1.28.1615
2836:(3): 260–264.
2818:
2786:
2760:
2745:
2713:
2694:
2664:
2663:
2661:
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2642:
2639:
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2164:
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2110:
2105:
2101:
2097:
2092:
2088:
2064:
2061:
2058:
2042:
2039:
2034:dot planimeter
2006:
2003:
1984:
1980:
1968:
1967:
1956:
1953:
1948:
1943:
1940:
1937:
1934:
1929:
1925:
1918:
1913:
1910:
1907:
1902:
1898:
1889:
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1519:
1516:
1512:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1473:floor function
1460:
1457:
1454:
1451:
1439:
1436:
1428:
1427:
1416:
1413:
1410:
1407:
1402:
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1197:
1193:
1187:
1183:
1179:
1175:
1171:
1168:
1165:
1162:
1157:
1153:
1149:
1145:
1140:
1136:
1133:
1129:
1125:
1122:
1119:
1116:
1112:
1090:
1089:
1078:
1075:
1072:
1067:
1062:
1059:
1055:
1051:
1048:
1045:
1042:
1038:
1011:
1008:
1005:
985:
982:
979:
976:
973:
970:
967:
947:
944:
941:
938:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
886:
881:
878:
875:
872:
869:
866:
861:
856:
853:
850:
847:
844:
841:
836:
831:
828:
808:
805:
802:
799:
796:
793:
773:
753:
750:
747:
744:
741:
738:
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715:
712:
709:
698:
697:
685:
682:
679:
676:
673:
668:
664:
660:
657:
654:
651:
648:
645:
620:
616:
612:
592:
566:
562:
558:
538:
535:
532:
529:
518:
515:
514:
513:
500:
472:
468:
464:
443:
422:
419:
416:
413:
393:
390:
387:
384:
375:is denoted by
364:
353:
352:
341:
336:
332:
328:
323:
319:
315:
310:
306:
272:
268:
264:
259:
255:
251:
246:
242:
213:
193:
173:
170:
167:
164:
161:
141:
138:
135:
113:
108:
94:
91:
70:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3041:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3011:
3009:
2999:
2998:
2993:
2988:
2983:
2982:
2977:
2974:
2969:
2968:
2964:
2955:
2951:
2947:
2943:
2939:
2935:
2931:
2927:
2926:
2918:
2916:
2914:
2910:
2906:(1–2): 24–29.
2905:
2902:(in Polish).
2901:
2894:
2890:
2884:
2881:
2876:
2869:
2866:
2861:
2857:
2853:
2849:
2844:
2839:
2835:
2831:
2830:
2822:
2819:
2814:
2810:
2806:
2805:
2800:
2796:
2790:
2787:
2782:
2778:
2774:
2770:
2769:Huxley, M. N.
2764:
2761:
2756:
2752:
2748:
2746:0-387-20860-7
2742:
2738:
2734:
2730:
2726:
2720:
2718:
2714:
2709:
2705:
2698:
2695:
2690:
2686:
2682:
2678:
2672:
2670:
2666:
2659:
2657:
2643:
2640:
2637:
2615:
2612:
2609:
2605:
2584:
2553:
2549:
2545:
2542:
2532:
2528:
2524:
2521:
2518:
2515:
2507:
2503:
2499:
2491:
2488:
2485:
2479:
2476:
2470:
2467:
2464:
2458:
2455:
2450:
2446:
2440:
2437:
2432:
2426:
2420:
2413:
2412:
2411:
2409:
2393:
2390:
2387:
2383:
2379:
2356:
2348:
2345:
2342:
2338:
2331:
2328:
2323:
2319:
2313:
2310:
2305:
2299:
2293:
2286:
2285:
2284:
2268:
2264:
2259:
2255:
2247:
2239:
2234:
2229:
2228:
2227:
2225:
2212:
2188:
2182:
2159:
2153:
2145:
2141:
2121:
2116:
2112:
2108:
2103:
2099:
2095:
2090:
2086:
2078:
2077:
2076:
2062:
2059:
2056:
2048:
2040:
2038:
2035:
2030:
2028:
2024:
2020:
2016:
2012:
2004:
2002:
2000:
1982:
1978:
1954:
1946:
1941:
1938:
1935:
1927:
1923:
1916:
1908:
1900:
1896:
1882:
1879:
1876:
1872:
1868:
1865:
1860:
1856:
1852:
1849:
1844:
1835:
1831:
1822:
1818:
1811:
1805:
1799:
1792:
1791:
1790:
1773:
1767:
1759:
1755:
1747:
1743:
1737:
1734:
1731:
1727:
1723:
1717:
1711:
1704:
1703:
1702:
1688:
1665:
1657:
1653:
1645:
1640:
1638:
1634:
1615:
1611:
1606:
1600:
1597:
1594:
1591:
1585:
1581:
1575:
1571:
1567:
1561:
1558:
1555:
1552:
1546:
1542:
1536:
1531:
1520:
1517:
1514:
1510:
1506:
1503:
1500:
1497:
1491:
1485:
1478:
1477:
1476:
1474:
1455:
1449:
1442:The value of
1437:
1435:
1433:
1432:Martin Huxley
1414:
1411:
1408:
1405:
1400:
1397:
1392:
1389:
1386:
1381:
1378:
1369:
1368:
1367:
1353:
1331:
1327:
1323:
1320:
1314:
1308:
1285:
1281:
1276:
1273:
1270:
1266:
1262:
1258:
1254:
1250:
1247:
1236:
1230:
1218:
1217:
1216:
1214:
1195:
1191:
1185:
1181:
1177:
1169:
1166:
1163:
1155:
1151:
1147:
1143:
1138:
1134:
1131:
1120:
1114:
1102:
1101:
1100:
1098:
1094:
1076:
1073:
1070:
1065:
1060:
1057:
1046:
1040:
1028:
1027:
1026:
1023:
1009:
1006:
1003:
983:
980:
977:
974:
971:
968:
965:
958:increases by
942:
936:
916:
913:
907:
901:
898:
895:
892:
884:
876:
873:
870:
867:
859:
851:
848:
845:
842:
834:
826:
806:
803:
797:
791:
771:
745:
739:
713:
707:
680:
674:
671:
666:
662:
658:
655:
649:
643:
636:
635:
634:
618:
614:
610:
590:
582:
564:
560:
556:
533:
527:
516:
511:
506:
501:
498:
493:
488:
487:
486:
470:
466:
462:
454:
441:
417:
411:
388:
382:
362:
339:
334:
330:
326:
321:
317:
313:
308:
304:
296:
295:
294:
292:
288:
270:
266:
262:
257:
253:
249:
244:
240:
231:
227:
211:
191:
168:
165:
162:
139:
136:
133:
111:
92:
90:
88:
84:
68:
61:
57:
53:
49:
45:
32:
19:
2995:
2979:
2932:(1): 69–81.
2929:
2923:
2903:
2899:
2883:
2874:
2868:
2833:
2827:
2821:
2803:
2789:
2772:
2763:
2728:
2707:
2703:
2697:
2680:
2677:Hardy, G. H.
2576:
2371:
2243:
2205:
2139:
2137:
2044:
2031:
2008:
1997:denotes the
1969:
1788:
1641:
1630:
1441:
1429:
1300:
1210:
1091:
1024:
699:
520:
434:
354:
290:
286:
96:
47:
41:
2997:3Blue1Brown
2795:Hilbert, D.
1438:Exact forms
549:is roughly
93:The problem
44:mathematics
3008:Categories
2710:: 263–283.
1211:using the
583:of radius
83:error term
2981:MathWorld
2954:119451320
2838:CiteSeerX
2638:ε
2616:ε
2613:−
2543:−
2525:
2519:
2489:
2477:−
2471:
2441:π
2394:ε
2349:ε
2314:π
2265:π
2109:≤
2019:hyperbola
2013:; indeed
1939:π
1888:∞
1873:∑
1853:π
1812:−
1728:∑
1572:−
1526:∞
1511:∑
1434:in 2000.
1412:…
1393:≤
1321:≤
1277:ε
1167:
1132:≠
1071:π
1058:≤
1007:π
752:∣
737:∣
659:π
611:π
557:π
463:π
327:≤
137:≥
2801:(1952).
2679:(1959).
2630:for any
1607:⌋
1576:⌊
1568:⌋
1537:⌊
1301:Writing
226:equation
3029:Circles
2946:1894296
2860:2589321
2813:0046650
2781:1956254
2755:2076335
2689:0106147
2236:in the
2233:A175341
2047:coprime
819:one has
508:in the
505:A075726
495:in the
492:A000328
2952:
2944:
2858:
2840:
2811:
2779:
2753:
2743:
2687:
2023:sphere
2011:conics
1970:where
1409:0.6298
1097:Landau
579:, the
184:where
60:radius
56:circle
46:, the
2950:S2CID
2896:(PDF)
2856:JSTOR
2660:Notes
1093:Hardy
2741:ISBN
2641:>
2238:OEIS
2204:for
2032:The
1387:<
1366:are
510:OEIS
497:OEIS
433:for
289:and
204:and
2934:doi
2930:135
2848:doi
2834:107
2733:doi
2522:log
2516:log
2486:log
2468:exp
2388:304
2380:221
2248:is
1401:208
1398:131
1164:log
917:81.
232:by
42:In
3010::
2994:,
2978:.
2948:.
2942:MR
2940:.
2928:.
2912:^
2898:.
2891:.
2854:.
2846:.
2832:.
2809:MR
2797:;
2777:MR
2751:MR
2749:.
2739:.
2716:^
2708:46
2706:.
2685:MR
2668:^
2240:).
1639:.
984:12
896:69
885:20
871:61
860:18
846:57
835:17
807:49
2984:.
2956:.
2936::
2904:2
2862:.
2850::
2815:.
2783:.
2757:.
2735::
2691:.
2644:0
2610:1
2606:r
2585:c
2562:)
2559:)
2554:5
2550:/
2546:1
2539:)
2533:2
2529:r
2513:(
2508:5
2504:/
2500:3
2496:)
2492:r
2483:(
2480:c
2474:(
2465:r
2462:(
2459:O
2456:+
2451:2
2447:r
2438:6
2433:=
2430:)
2427:r
2424:(
2421:V
2391:+
2384:/
2357:.
2354:)
2346:+
2343:1
2339:r
2335:(
2332:O
2329:+
2324:2
2320:r
2311:6
2306:=
2303:)
2300:r
2297:(
2294:V
2269:2
2260:/
2256:6
2213:r
2192:)
2189:r
2186:(
2183:V
2163:)
2160:r
2157:(
2154:V
2122:.
2117:2
2113:r
2104:2
2100:n
2096:+
2091:2
2087:m
2063:n
2060:,
2057:m
1983:1
1979:J
1955:,
1952:)
1947:n
1942:x
1936:2
1933:(
1928:1
1924:J
1917:n
1912:)
1909:n
1906:(
1901:2
1897:r
1883:1
1880:=
1877:n
1869:x
1866:+
1861:2
1857:x
1850:=
1845:2
1841:)
1836:2
1832:x
1828:(
1823:2
1819:r
1809:)
1806:x
1803:(
1800:N
1774:.
1771:)
1768:n
1765:(
1760:2
1756:r
1748:2
1744:r
1738:0
1735:=
1732:n
1724:=
1721:)
1718:r
1715:(
1712:N
1689:n
1669:)
1666:n
1663:(
1658:2
1654:r
1616:.
1612:)
1601:3
1598:+
1595:i
1592:4
1586:2
1582:r
1562:1
1559:+
1556:i
1553:4
1547:2
1543:r
1532:(
1521:0
1518:=
1515:i
1507:4
1504:+
1501:1
1498:=
1495:)
1492:r
1489:(
1486:N
1459:)
1456:r
1453:(
1450:N
1415:,
1406:=
1390:t
1382:2
1379:1
1354:t
1332:t
1328:r
1324:C
1318:)
1315:r
1312:(
1309:E
1286:.
1282:)
1274:+
1271:2
1267:/
1263:1
1259:r
1255:(
1251:O
1248:=
1244:|
1240:)
1237:r
1234:(
1231:E
1227:|
1196:,
1192:)
1186:4
1182:/
1178:1
1174:)
1170:r
1161:(
1156:2
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1148:1
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1139:(
1135:o
1128:|
1124:)
1121:r
1118:(
1115:E
1111:|
1077:.
1074:r
1066:2
1061:2
1054:|
1050:)
1047:r
1044:(
1041:E
1037:|
1010:r
1004:2
981:,
978:8
975:,
972:4
969:,
966:8
946:)
943:r
940:(
937:E
914:=
911:)
908:5
905:(
902:N
899:,
893:=
890:)
880:(
877:N
874:,
868:=
865:)
855:(
852:N
849:,
843:=
840:)
830:(
827:N
804:=
801:)
798:4
795:(
792:N
772:r
749:)
746:r
743:(
740:E
717:)
714:r
711:(
708:E
684:)
681:r
678:(
675:E
672:+
667:2
663:r
656:=
653:)
650:r
647:(
644:N
619:2
615:r
591:r
565:2
561:r
537:)
534:r
531:(
528:N
512:)
499:)
471:2
467:r
442:r
421:)
418:r
415:(
412:N
392:)
389:r
386:(
383:N
363:r
340:.
335:2
331:r
322:2
318:n
314:+
309:2
305:m
291:n
287:m
271:2
267:r
263:=
258:2
254:y
250:+
245:2
241:x
212:n
192:m
172:)
169:n
166:,
163:m
160:(
140:0
134:r
112:2
107:R
69:r
36:π
20:)
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