597:
446:
203:
477:
314:
87:
261:
228:
by
Montgomery and Vaughan; an earlier result of Brun and Titchmarsh obtained a weaker version of this inequality with an additional multiplicative factor of
631:
468:
455:, discovered by himself. Later this idea of exploiting structures in sieving errors developed into a major method in Analytic Number Theory, due to
322:
740:
115:
695:
745:
664:
643:
37:
691:
684:
674:
592:{\displaystyle \pi (x;q,a)={\frac {x}{\varphi (q)\log(x)}}\left({1+O\left({\frac {1}{\log x}}\right)}\right)}
679:
615:
29:
278:
51:
17:
231:
451:
This is due to Y. Motohashi (1973). He used a bilinear structure in the error term in the
8:
652:
660:
639:
717:
709:
456:
713:
734:
452:
441:{\displaystyle \pi (x;q,a)\leq {(2+o(1))x \over \varphi (q)\log(x/q^{3/8})}}
225:
722:
700:
33:
25:
602:
but this can only be proved to hold for the more restricted range
471:
gives an asymptotic result, which may be expressed in the form
198:{\displaystyle \pi (x;q,a)\leq {2x \over \varphi (q)\log(x/q)}}
657:
Applications of sieve methods to the theory of numbers
462:
480:
325:
281:
234:
118:
54:
591:
440:
308:
255:
197:
81:
690:
732:
469:Dirichlet's theorem on arithmetic progressions
659:, Cambridge University Press, p. 10,
721:
630:
38:prime numbers in arithmetic progression
733:
672:
651:
636:Sieve Methods and Prime Number Theory
459:'s extension to combinatorial sieve.
316:, then there exists a better bound:
463:Comparison with Dirichlet's theorem
13:
741:Theorems in analytic number theory
14:
757:
638:, Tata IFR and Springer-Verlag,
266:
538:
532:
523:
517:
502:
484:
432:
403:
394:
388:
377:
374:
368:
356:
347:
329:
309:{\displaystyle q\leq x^{9/20}}
250:
244:
189:
175:
166:
160:
140:
122:
76:
58:
1:
624:
746:Theorems about prime numbers
43:
7:
698:(1973), "The large sieve",
680:Encyclopedia of Mathematics
275:is relatively small, e.g.,
89:count the number of primes
82:{\displaystyle \pi (x;q,a)}
10:
762:
606: < (log
219:
714:10.1112/s0025579300004708
675:"Brun-Titchmarsh theorem"
224:The result was proven by
30:Edward Charles Titchmarsh
36:on the distribution of
22:Brun–Titchmarsh theorem
616:Siegel–Walfisz theorem
593:
442:
310:
257:
256:{\displaystyle 1+o(1)}
199:
83:
18:analytic number theory
594:
443:
311:
258:
200:
84:
673:Mikawa, H. (2001) ,
478:
323:
279:
232:
116:
52:
653:Hooley, Christopher
589:
438:
306:
253:
195:
79:
632:Motohashi, Yoichi
578:
542:
436:
193:
753:
726:
725:
692:Montgomery, H.L.
687:
669:
648:
598:
596:
595:
590:
588:
584:
583:
579:
577:
563:
543:
541:
509:
447:
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431:
430:
426:
413:
383:
354:
315:
313:
312:
307:
305:
304:
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262:
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259:
254:
212: <
204:
202:
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196:
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185:
155:
147:
88:
86:
85:
80:
761:
760:
756:
755:
754:
752:
751:
750:
731:
730:
667:
646:
627:
621:
610:) for constant
567:
562:
558:
548:
544:
513:
508:
479:
476:
475:
465:
422:
418:
414:
409:
384:
355:
353:
324:
321:
320:
296:
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288:
280:
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276:
269:
233:
230:
229:
222:
181:
156:
148:
146:
117:
114:
113:
53:
50:
49:
46:
12:
11:
5:
759:
749:
748:
743:
729:
728:
723:2027.42/152543
708:(2): 119–134,
688:
670:
665:
649:
644:
626:
623:
614:: this is the
600:
599:
587:
582:
576:
573:
570:
566:
561:
557:
554:
551:
547:
540:
537:
534:
531:
528:
525:
522:
519:
516:
512:
507:
504:
501:
498:
495:
492:
489:
486:
483:
464:
461:
449:
448:
434:
429:
425:
421:
417:
412:
408:
405:
402:
399:
396:
393:
390:
387:
382:
379:
376:
373:
370:
367:
364:
361:
358:
352:
349:
346:
343:
340:
337:
334:
331:
328:
303:
299:
295:
291:
287:
284:
268:
265:
252:
249:
246:
243:
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237:
221:
218:
206:
205:
191:
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184:
180:
177:
174:
171:
168:
165:
162:
159:
154:
151:
145:
142:
139:
136:
133:
130:
127:
124:
121:
78:
75:
72:
69:
66:
63:
60:
57:
45:
42:
24:, named after
9:
6:
4:
3:
2:
758:
747:
744:
742:
739:
738:
736:
724:
719:
715:
711:
707:
703:
702:
697:
696:Vaughan, R.C.
693:
689:
686:
682:
681:
676:
671:
668:
666:0-521-20915-3
662:
658:
654:
650:
647:
645:3-540-12281-8
641:
637:
633:
629:
628:
622:
619:
617:
613:
609:
605:
585:
580:
574:
571:
568:
564:
559:
555:
552:
549:
545:
535:
529:
526:
520:
514:
510:
505:
499:
496:
493:
490:
487:
481:
474:
473:
472:
470:
467:By contrast,
460:
458:
454:
453:Selberg sieve
427:
423:
419:
415:
410:
406:
400:
397:
391:
385:
380:
371:
365:
362:
359:
350:
344:
341:
338:
335:
332:
326:
319:
318:
317:
301:
297:
293:
289:
285:
282:
274:
264:
247:
241:
238:
235:
227:
226:sieve methods
217:
215:
211:
186:
182:
178:
172:
169:
163:
157:
152:
149:
143:
137:
134:
131:
128:
125:
119:
112:
111:
110:
108:
105: ≤
104:
100:
96:
93:congruent to
92:
73:
70:
67:
64:
61:
55:
41:
39:
35:
31:
27:
23:
19:
705:
699:
678:
656:
635:
620:
611:
607:
603:
601:
466:
450:
272:
270:
267:Improvements
223:
213:
209:
207:
106:
102:
98:
97:modulo
94:
90:
47:
21:
15:
701:Mathematika
34:upper bound
735:Categories
625:References
457:H. Iwaniec
26:Viggo Brun
685:EMS Press
572:
530:
515:φ
482:π
401:
386:φ
351:≤
327:π
286:≤
173:
158:φ
144:≤
120:π
56:π
44:Statement
655:(1976),
634:(1983),
208:for all
32:, is an
220:History
109:. Then
663:
642:
20:, the
101:with
661:ISBN
640:ISBN
48:Let
40:.
28:and
718:hdl
710:doi
569:log
527:log
398:log
271:If
170:log
16:In
737::
716:,
706:20
704:,
694:;
683:,
677:,
618:.
302:20
263:.
216:.
727:.
720::
712::
612:c
608:x
604:q
586:)
581:)
575:x
565:1
560:(
556:O
553:+
550:1
546:(
539:)
536:x
533:(
524:)
521:q
518:(
511:x
506:=
503:)
500:a
497:,
494:q
491:;
488:x
485:(
433:)
428:8
424:/
420:3
416:q
411:/
407:x
404:(
395:)
392:q
389:(
381:x
378:)
375:)
372:1
369:(
366:o
363:+
360:2
357:(
348:)
345:a
342:,
339:q
336:;
333:x
330:(
298:/
294:9
290:x
283:q
273:q
251:)
248:1
245:(
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239:+
236:1
214:x
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190:)
187:q
183:/
179:x
176:(
167:)
164:q
161:(
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150:2
141:)
138:a
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129:;
126:x
123:(
107:x
103:p
99:q
95:a
91:p
77:)
74:a
71:,
68:q
65:;
62:x
59:(
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