633:. These two bounds are related, as we shall see, and come from fairly elementary considerations. In the series of papers Voronoy developed geometric and analytic methods to improve both Dirichlet’s and Gauss’ bound. Most importantly in retrospect, he generalized the formula by allowing weighted sums, at the expense of introducing more general integral operations on f than the
59:
To
Voronoy and his contemporaries, the formula appeared tailor-made to evaluate certain finite sums. That seemed significant because several important questions in number theory involve finite sums of arithmetic quantities. In this connection, let us mention two classical examples,
846:
362:
219:
486:
631:
550:
939:(1904). Sur une fonction transcendente et ses applications à la sommation de quelques séries. In Annales Scientifiques de l'École Normale Supérieure (Vol. 21, pp. 207–267).
878:
579:
242:
702:
254:
933:
Miller, S. D., & Schmid, W. (2006). Automorphic distributions, L-functions, and
Voronoi summation for GL(3). Annals of mathematics, 423–488.
90:
377:
43:. The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their
957:
61:
47:-functions. There have been numerous results coming out the Voronoi formula on GL(2). The concept is named after
952:
36:
584:
502:
881:
32:
65:
245:
854:
555:
634:
40:
227:
923:
893:
28:
650:
841:{\displaystyle \sum _{n}a(n)e(an/c)\omega (n)=\sum _{n}a(n)e(-{\bar {a}}n/c)\Omega (n),}
936:
48:
24:
946:
654:
81:
357:{\displaystyle r_{2}(n)=\#\{(x,y)\in \mathbb {Z} ^{2}\mid x^{2}+y^{2}=n\},}
214:{\displaystyle D(X)=\sum _{n=1}^{X}d(n)-X\log X-(2\gamma -1)X=O(X^{1/2})}
927:
914:
Good, Anton (1984), "Cusp forms and eigenfunctions of the
Laplacian",
248:≈ 0.57721566. Gauss’ circle problem concerns the average size of
481:{\displaystyle \Delta (X)=\sum _{n=1}^{X}r_{2}(n)-\pi X=O(X^{1/2}).}
368:
692:
be a well-behaved test function. The
Voronoi formula for
76:), the number of positive divisors of an integer
857:
705:
587:
558:
505:
380:
257:
230:
93:
872:
840:
625:
573:
544:
491:Each problem has a geometric interpretation, with
480:
356:
236:
213:
8:
620:
588:
539:
506:
348:
283:
859:
858:
856:
812:
798:
797:
770:
740:
710:
704:
608:
595:
586:
557:
504:
462:
458:
421:
411:
400:
379:
336:
323:
310:
306:
305:
262:
256:
229:
198:
194:
124:
113:
92:
16:Mathematical formula in harmonic analysis
499:) counting lattice points in the region
35:on either side. It can be regarded as a
626:{\displaystyle \{x^{2}+y^{2}\leq X\}}
545:{\displaystyle \{x,y>0,xy\leq X\}}
7:
901:
68:. The former estimates the size of
31:, with the coefficients twisted by
823:
559:
381:
280:
14:
672:) its Fourier coefficients. Let
864:
832:
826:
820:
803:
791:
785:
779:
760:
754:
748:
731:
725:
719:
568:
562:
472:
451:
433:
427:
390:
384:
298:
286:
274:
268:
208:
187:
175:
160:
139:
133:
103:
97:
1:
892:and Ω is a certain integral
581:lattice points in the disc
62:Dirichlet's divisor problem
974:
873:{\displaystyle {\bar {a}}}
574:{\displaystyle \Delta (X)}
37:Poisson summation formula
23:is an equality involving
237:{\displaystyle \gamma }
958:Analytic number theory
882:multiplicative inverse
874:
842:
627:
575:
546:
482:
416:
358:
238:
215:
129:
916:Mathematische Annalen
875:
843:
628:
576:
547:
483:
396:
359:
239:
216:
109:
55:Classical application
855:
703:
585:
556:
503:
378:
255:
228:
91:
66:Gauss circle problem
25:Fourier coefficients
33:additive characters
928:10.1007/bf01451932
870:
838:
775:
715:
680:be integers with (
641:Modern formulation
623:
571:
542:
478:
371:gave the estimate
354:
234:
211:
41:non-abelian groups
19:In mathematics, a
953:Automorphic forms
867:
806:
766:
706:
635:Fourier transform
29:automorphic forms
965:
930:
894:Hankel transform
879:
877:
876:
871:
869:
868:
860:
847:
845:
844:
839:
816:
808:
807:
799:
774:
744:
714:
632:
630:
629:
624:
613:
612:
600:
599:
580:
578:
577:
572:
551:
549:
548:
543:
487:
485:
484:
479:
471:
470:
466:
426:
425:
415:
410:
363:
361:
360:
355:
341:
340:
328:
327:
315:
314:
309:
267:
266:
246:Euler's constant
243:
241:
240:
235:
220:
218:
217:
212:
207:
206:
202:
128:
123:
973:
972:
968:
967:
966:
964:
963:
962:
943:
942:
913:
910:
853:
852:
701:
700:
651:Maass cusp form
643:
604:
591:
583:
582:
554:
553:
501:
500:
454:
417:
376:
375:
332:
319:
304:
258:
253:
252:
226:
225:
190:
89:
88:
57:
21:Voronoi formula
17:
12:
11:
5:
971:
969:
961:
960:
955:
945:
944:
941:
940:
934:
931:
922:(4): 523–548,
909:
906:
866:
863:
849:
848:
837:
834:
831:
828:
825:
822:
819:
815:
811:
805:
802:
796:
793:
790:
787:
784:
781:
778:
773:
769:
765:
762:
759:
756:
753:
750:
747:
743:
739:
736:
733:
730:
727:
724:
721:
718:
713:
709:
642:
639:
622:
619:
616:
611:
607:
603:
598:
594:
590:
570:
567:
564:
561:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
489:
488:
477:
474:
469:
465:
461:
457:
453:
450:
447:
444:
441:
438:
435:
432:
429:
424:
420:
414:
409:
406:
403:
399:
395:
392:
389:
386:
383:
365:
364:
353:
350:
347:
344:
339:
335:
331:
326:
322:
318:
313:
308:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
265:
261:
233:
222:
221:
210:
205:
201:
197:
193:
189:
186:
183:
180:
177:
174:
171:
168:
165:
162:
159:
156:
153:
150:
147:
144:
141:
138:
135:
132:
127:
122:
119:
116:
112:
108:
105:
102:
99:
96:
56:
53:
49:Georgy Voronoy
15:
13:
10:
9:
6:
4:
3:
2:
970:
959:
956:
954:
951:
950:
948:
938:
935:
932:
929:
925:
921:
917:
912:
911:
907:
905:
903:
899:
895:
891:
887:
883:
861:
835:
829:
817:
813:
809:
800:
794:
788:
782:
776:
771:
767:
763:
757:
751:
745:
741:
737:
734:
728:
722:
716:
711:
707:
699:
698:
697:
695:
691:
687:
683:
679:
675:
671:
667:
663:
659:
656:
655:modular group
652:
648:
640:
638:
636:
617:
614:
609:
605:
601:
596:
592:
565:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
498:
494:
475:
467:
463:
459:
455:
448:
445:
442:
439:
436:
430:
422:
418:
412:
407:
404:
401:
397:
393:
387:
374:
373:
372:
370:
351:
345:
342:
337:
333:
329:
324:
320:
316:
311:
301:
295:
292:
289:
277:
271:
263:
259:
251:
250:
249:
247:
231:
203:
199:
195:
191:
184:
181:
178:
172:
169:
166:
163:
157:
154:
151:
148:
145:
142:
136:
130:
125:
120:
117:
114:
110:
106:
100:
94:
87:
86:
85:
83:
79:
75:
71:
67:
63:
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
919:
915:
897:
889:
888:modulo
885:
850:
693:
689:
685:
681:
677:
673:
669:
665:
661:
657:
646:
644:
496:
492:
490:
366:
223:
77:
73:
69:
58:
44:
20:
18:
937:Voronoï, G.
902:Good (1984)
688:) = 1. Let
947:Categories
908:References
367:for which
865:¯
824:Ω
804:¯
795:−
768:∑
752:ω
708:∑
615:≤
560:Δ
534:≤
440:π
437:−
398:∑
382:Δ
317:∣
302:∈
281:#
232:γ
170:−
167:γ
158:−
152:
143:−
111:∑
82:Dirichlet
896:of
653:for the
64:and the
900:. (see
696:states
84:proved
851:where
664:) and
552:, and
224:where
880:is a
649:be a
369:Gauss
645:Let
519:>
39:for
924:doi
920:255
884:of
660:(2,
658:PSL
244:is
149:log
27:of
949::
918:,
904:)
637:.
80:.
51:.
926::
898:ω
890:c
886:a
862:a
836:,
833:)
830:n
827:(
821:)
818:c
814:/
810:n
801:a
792:(
789:e
786:)
783:n
780:(
777:a
772:n
764:=
761:)
758:n
755:(
749:)
746:c
742:/
738:n
735:a
732:(
729:e
726:)
723:n
720:(
717:a
712:n
694:ƒ
690:ω
686:c
684:,
682:a
678:c
676:,
674:a
670:n
668:(
666:a
662:Z
647:ƒ
621:}
618:X
610:2
606:y
602:+
597:2
593:x
589:{
569:)
566:X
563:(
540:}
537:X
531:y
528:x
525:,
522:0
516:y
513:,
510:x
507:{
497:X
495:(
493:D
476:.
473:)
468:2
464:/
460:1
456:X
452:(
449:O
446:=
443:X
434:)
431:n
428:(
423:2
419:r
413:X
408:1
405:=
402:n
394:=
391:)
388:X
385:(
352:,
349:}
346:n
343:=
338:2
334:y
330:+
325:2
321:x
312:2
307:Z
299:)
296:y
293:,
290:x
287:(
284:{
278:=
275:)
272:n
269:(
264:2
260:r
209:)
204:2
200:/
196:1
192:X
188:(
185:O
182:=
179:X
176:)
173:1
164:2
161:(
155:X
146:X
140:)
137:n
134:(
131:d
126:X
121:1
118:=
115:n
107:=
104:)
101:X
98:(
95:D
78:n
74:n
72:(
70:d
45:L
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