958:
84:
74:
53:
570:
1077:
other, but that the sum over n mod c could give a wrong identity--at least I don't see why it should be correct. The second factor in the sum has no sawtooth function around the n, so that factor is not independent on which residue system we're summing over. (It could very well happen that the sum over n=0..c-1 gives the same result as a sum over n=c..2c-1, but those are not the only complete residue systems we may choose...)
22:
953:{\displaystyle \sum _{n=0}^{c-1}\left(\left({\frac {bn}{c}}\right)\right)\left({\frac {n}{c}}+x\right)=\sum _{n=0}^{c-1}\left(\left({\frac {bn}{c}}\right)\right)\left({\frac {n}{c}}-{\frac {1}{2}}\right)+\sum _{n=0}^{c-1}\left(\left({\frac {bn}{c}}\right)\right)\left(x+{\frac {1}{2}}\right)=s(b,c)+\left(x+{\frac {1}{2}}\right)\sum _{n=0}^{c-1}\left(\left({\frac {bn}{c}}\right)\right),}
526:
1076:
I see where the confusion was and I'm glad that's clear now. (The literature is also confusing regarding this notation--and I'm one of the culprits who used different notations in different papers...) However, the summation condition is still an issue. I'm not claiming that one is more valid than the
1096:
As stated, the "formula" is wrong, no matter whether b and c are relatively prime or not. Plug in b=1, c=4, x=0. Take the sum on the right-hand side over n=0,1,2,3, and then over n=0,1,6,7. Both are complete residue systems mod 4, but they give different sums. We can fix things by demanding that n
972:
Argh. We are both confused and going in circles. First, I mislead you and myself, since I use a definition of ((x)) that differs from the article (I take it to be -1/2 for integers, and not 0. I do this for numerical exploration, and I am now realizing that this is actually a subtle point. Among
332:
One of the many roles a WP math article can take is as a reference, which can include formulas that, while "obvious" to the specialist, may not be obvious to the casual reader. Think of a table of integrals: A lot of the integrals listed in a table, or even many of the formulas in
286:
I don't think the "another useful formula" (at the end of "simple formulae") is correct--for once, the second factor in the sum clearly depends on which residue class we're summing over (which it shouldn't). If we change the summation condition to n running from 1 to
352:
I outlined a proof that the "another useful formula" was wrong: as stated, the formula violated the fact that s(a+b,b) = s(a,b). I took it out. By the way, I couldn't agree more to your second comment, and it is for these reasons that I despise the word "obvious".
322:
If I added this formula (I don't remember), I would have copied it from a reputable source, e.g. one of Tom
Apostol's books, and so don't be too quick to assume there's an error. (There may be one, but don't assume it unless you can prove
376:
560:
Hmmm... I think there are similar problems as before. You'll get different answers for the sum on the right, depending on how you let n vary (e.g., you get two different answers as n ranges from 0 to c-1 or from c to
1085:
Sorry, guys. I forgot the condition (b,c)=1. I reinstated the formula, but if you think it's not worth being mention because it indeed is just a simple corollary of the formula above, feel free to remove it.
206:
This formula is cited in the Beck and
Robbins PDF. Do you believe its wrong? Just right now, I can't think of any quick and obvious way to verify this. Anyway, we should have this conversation on
140:
337:
are "obvious" when you've been doing it for a while. However, "another useful formula" doesn't seem at all to be immediately obvious, so keeping it seems like a good idea to me.
1025:
1064:
196:
for s(b,c) involving cots is correct for the saw-tooth function as now (and previously) defined on the page. I've corrected the other alternative form of s(b,c) myself.
315:
Hmm. Please make the correction. I am not studying this stuff at this time, so I cannot just tell be looking at it whether its correct or not. However, several remarks:
220:
Also, not sure what you mean by "I corrected the other formula"; I am a bit lazy in trying to decipher the changelog. What was the problem, and how was it corrected?
521:{\displaystyle s(b,c)-p{\frac {x}{2}}=\sum _{n\mod c}\left(\left({\frac {bn}{c}}\right)\right)\left({\frac {n}{c}}+x\right),\qquad \forall x\in \mathbb {R} .}
1119:
130:
1114:
976:
Second, using the article definition of ((x)), you have just proved that the original formula was correct, i.e. that the result does not depend on
106:
97:
58:
1066:, it seems that both give the same answers. That's kind of the point of having a sawtooth, to get rid of the integer part ...
33:
291:-1, then the formula is correct. But beyond that, I don't quite see its usefulness: one can distribute to separate
1097:
runs from 0 to c-1, but then the "formula" is a simple corollary of the formula right before, as shown above.
21:
252:
I corrected the "Aternate forms" section--it was lacking the assumptions of coprimeness, for both formulas.
235:
Well, the formula stayed actually the same, just the following comment was changed: omega != 1, of course.
334:
181:
39:
83:
986:
1030:
307:
plays no role. I suggest taking the "another useful formula" out. (But not the "Furthermore..."!)
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
172:
Sorry about the big mod below the sum but I don't know how to make it smaller. Any suggestions?
73:
52:
1087:
364:
268:
236:
197:
565:
But even if we agree on letting n range from 0 to c-1, I don't see your formula... then
964:
and this last sum on the right is zero, independent of the relationship of b and c...
1108:
1098:
1078:
965:
354:
308:
253:
363:
OK, this time, I'm putting my brain in gear. The incorrect formula(s) were added by
303:, and then a quick application of the identity just before this one shows that the
193:
177:
102:
550:
18:19, 1 December 2005 (UTC) -- it seems to be gcd at least for positve b,c --
1067:
551:
547:
338:
221:
210:
79:
192:
Could you please check if the formula you added under
Alternative forms on
539:
so that p=gcd(b,c). So in some way it is kind-of an interesting formula,
1090:
1070:
554:
341:
271:
239:
224:
213:
200:
15:
367:
on 21 November 2005. We need to ask him to explain himself.
973:
other things, it leads to that curious gcd identity).
1033:
989:
573:
379:
101:, a collaborative effort to improve the coverage of
1058:
1019:
952:
520:
983:Third, I am not sure why you are insisting that
8:
19:
47:
1038:
1032:
1005:
994:
988:
924:
904:
893:
874:
824:
790:
770:
759:
737:
724:
696:
676:
665:
637:
609:
589:
578:
572:
511:
510:
475:
447:
421:
404:
378:
432:
430:
267:Sorry, wasn't quite alert as it seems.
49:
7:
95:This article is within the scope of
1027:is somehow more valid than the sum
537:in a way that I can't quite make ou
370:In the meanwhile, I'm finding that
38:It is of interest to the following
501:
14:
1120:Low-priority mathematics articles
1020:{\displaystyle \sum _{n=0}^{c-1}}
425:
115:Knowledge:WikiProject Mathematics
1115:Start-Class mathematics articles
180:is to a totally unrelated page.
118:Template:WikiProject Mathematics
82:
72:
51:
20:
135:This article has been rated as
857:
845:
395:
383:
1:
1059:{\displaystyle \sum _{nmodc}}
499:
225:20:18, 25 November 2005 (UTC)
214:20:15, 25 November 2005 (UTC)
201:18:42, 25 November 2005 (UTC)
109:and see a list of open tasks.
1091:21:06, 2 December 2005 (UTC)
1071:21:14, 1 December 2005 (UTC)
555:18:33, 1 December 2005 (UTC)
342:00:04, 1 December 2005 (UTC)
272:21:00, 2 December 2005 (UTC)
240:21:00, 2 December 2005 (UTC)
1136:
184:10:31, 11 Feb 2005 (UTC)
134:
67:
46:
541:esp. If I can ping down
282:"Another useful formula"
141:project's priority scale
98:WikiProject Mathematics
1060:
1021:
1016:
954:
915:
781:
687:
600:
522:
28:This article is rated
1061:
1022:
990:
955:
889:
755:
661:
574:
523:
335:Abramowitz and Stegun
1031:
987:
571:
535:that depends on b,c
377:
121:mathematics articles
1056:
1055:
1017:
950:
518:
500:
438:
433:
431:
426:
176:The 'see also' to
90:Mathematics portal
34:content assessment
1034:
937:
882:
832:
803:
745:
732:
709:
645:
622:
531:for some integer
483:
460:
417:
412:
311:29 November 2005
208:Talk:Dedekind sum
188:cotangent formula
155:
154:
151:
150:
147:
146:
1127:
1065:
1063:
1062:
1057:
1054:
1026:
1024:
1023:
1018:
1015:
1004:
959:
957:
956:
951:
946:
942:
938:
933:
925:
914:
903:
888:
884:
883:
875:
838:
834:
833:
825:
812:
808:
804:
799:
791:
780:
769:
751:
747:
746:
738:
733:
725:
718:
714:
710:
705:
697:
686:
675:
657:
653:
646:
638:
631:
627:
623:
618:
610:
599:
588:
527:
525:
524:
519:
514:
495:
491:
484:
476:
469:
465:
461:
456:
448:
437:
413:
405:
256:29 November 2005
182:Charles Matthews
169:
168:
164:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1135:
1134:
1130:
1129:
1128:
1126:
1125:
1124:
1105:
1104:
1101:2 December 2005
1081:1 December 2005
1029:
1028:
985:
984:
968:1 December 2005
926:
920:
916:
867:
863:
817:
813:
792:
786:
782:
723:
719:
698:
692:
688:
636:
632:
611:
605:
601:
569:
568:
474:
470:
449:
443:
439:
375:
374:
357:1 December 2005
284:
190:
170:
166:
162:
160:
159:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1133:
1131:
1123:
1122:
1117:
1107:
1106:
1103:
1102:
1083:
1082:
1053:
1050:
1047:
1044:
1041:
1037:
1014:
1011:
1008:
1003:
1000:
997:
993:
970:
969:
962:
961:
960:
949:
945:
941:
936:
932:
929:
923:
919:
913:
910:
907:
902:
899:
896:
892:
887:
881:
878:
873:
870:
866:
862:
859:
856:
853:
850:
847:
844:
841:
837:
831:
828:
823:
820:
816:
811:
807:
802:
798:
795:
789:
785:
779:
776:
773:
768:
765:
762:
758:
754:
750:
744:
741:
736:
731:
728:
722:
717:
713:
708:
704:
701:
695:
691:
685:
682:
679:
674:
671:
668:
664:
660:
656:
652:
649:
644:
641:
635:
630:
626:
621:
617:
614:
608:
604:
598:
595:
592:
587:
584:
581:
577:
563:
562:
529:
528:
517:
513:
509:
506:
503:
498:
494:
490:
487:
482:
479:
473:
468:
464:
459:
455:
452:
446:
442:
436:
429:
424:
420:
416:
411:
408:
403:
400:
397:
394:
391:
388:
385:
382:
361:
360:
359:
358:
347:
346:
345:
344:
327:
326:
325:
324:
317:
316:
283:
280:
279:
278:
277:
276:
275:
274:
260:
259:
258:
257:
247:
246:
245:
244:
243:
242:
228:
227:
217:
216:
189:
186:
174:
158:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1132:
1121:
1118:
1116:
1113:
1112:
1110:
1100:
1095:
1094:
1093:
1092:
1089:
1080:
1075:
1074:
1073:
1072:
1069:
1051:
1048:
1045:
1042:
1039:
1035:
1012:
1009:
1006:
1001:
998:
995:
991:
981:
979:
974:
967:
963:
947:
943:
939:
934:
930:
927:
921:
917:
911:
908:
905:
900:
897:
894:
890:
885:
879:
876:
871:
868:
864:
860:
854:
851:
848:
842:
839:
835:
829:
826:
821:
818:
814:
809:
805:
800:
796:
793:
787:
783:
777:
774:
771:
766:
763:
760:
756:
752:
748:
742:
739:
734:
729:
726:
720:
715:
711:
706:
702:
699:
693:
689:
683:
680:
677:
672:
669:
666:
662:
658:
654:
650:
647:
642:
639:
633:
628:
624:
619:
615:
612:
606:
602:
596:
593:
590:
585:
582:
579:
575:
567:
566:
564:
559:
558:
557:
556:
553:
549:
545:
544:
538:
534:
515:
507:
504:
496:
492:
488:
485:
480:
477:
471:
466:
462:
457:
453:
450:
444:
440:
434:
427:
422:
418:
414:
409:
406:
401:
398:
392:
389:
386:
380:
373:
372:
371:
368:
366:
356:
351:
350:
349:
348:
343:
340:
336:
331:
330:
329:
328:
321:
320:
319:
318:
314:
313:
312:
310:
306:
302:
298:
294:
290:
281:
273:
270:
266:
265:
264:
263:
262:
261:
255:
251:
250:
249:
248:
241:
238:
234:
233:
232:
231:
230:
229:
226:
223:
219:
218:
215:
212:
209:
205:
204:
203:
202:
199:
195:
187:
185:
183:
179:
173:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1084:
982:
977:
975:
971:
542:
540:
536:
532:
530:
369:
362:
304:
300:
296:
292:
288:
285:
207:
194:Dedekind sum
191:
178:Dedekind cut
175:
171:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1109:Categories
365:User:Ncik
1099:mattbeck
1079:mattbeck
966:mattbeck
355:mattbeck
309:mattbeck
254:mattbeck
139:on the
561:2c-1).
161:": -->
36:scale.
1068:linas
552:linas
548:linas
339:linas
222:linas
211:linas
1088:Ncik
323:it).
299:and
269:Ncik
237:Ncik
198:Ncik
163:edit
428:mod
131:Low
1111::
1036:∑
1010:−
992:∑
980:.
909:−
891:∑
775:−
757:∑
735:−
681:−
663:∑
594:−
576:∑
546:.
508:∈
502:∀
419:∑
399:−
1052:c
1049:d
1046:o
1043:m
1040:n
1013:1
1007:c
1002:0
999:=
996:n
978:x
948:,
944:)
940:)
935:c
931:n
928:b
922:(
918:(
912:1
906:c
901:0
898:=
895:n
886:)
880:2
877:1
872:+
869:x
865:(
861:+
858:)
855:c
852:,
849:b
846:(
843:s
840:=
836:)
830:2
827:1
822:+
819:x
815:(
810:)
806:)
801:c
797:n
794:b
788:(
784:(
778:1
772:c
767:0
764:=
761:n
753:+
749:)
743:2
740:1
730:c
727:n
721:(
716:)
712:)
707:c
703:n
700:b
694:(
690:(
684:1
678:c
673:0
670:=
667:n
659:=
655:)
651:x
648:+
643:c
640:n
634:(
629:)
625:)
620:c
616:n
613:b
607:(
603:(
597:1
591:c
586:0
583:=
580:n
543:p
533:p
516:.
512:R
505:x
497:,
493:)
489:x
486:+
481:c
478:n
472:(
467:)
463:)
458:c
454:n
451:b
445:(
441:(
435:c
423:n
415:=
410:2
407:x
402:p
396:)
393:c
390:,
387:b
384:(
381:s
305:x
301:x
297:c
295:/
293:n
289:c
167:]
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
