92:
61:
671:
511:
702:
Being evaluated to such a simple number has made the term
Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.
569:
267:
420:
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859:
293:
332:
199:
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882:« Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183–256 et 281–361
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764:. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974
954:
1141:
1126:
1046:
986:
1021:
516:
Not only is it now known that the limit exists, but also that its value is equal to 1, somewhat less than
Legendre's
1095:
971:
666:{\displaystyle \pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }
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1026:
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indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by
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21:
This article uses technical mathematical notation for logarithms. All instances of
863:, Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220
801:
699:
and La Vallée
Poussin, but without any estimate of the involved error term).
506:{\displaystyle B=\lim _{n\to \infty }\left(\log(n)-{n \over \pi (n)}\right),}
905:
914:
557:
exists, it must be equal to 1. An easier proof was given by Pintz in 1980.
822:
La Vallée
Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899
809:
775:
87:) (red line) appear to converge to a value around 1.08366 (blue line).
564:, under the precise form with an explicit estimate of the error term
118:) (red line) appear to be consistently less than 1.08366 (blue line).
793:
90:
59:
918:
523:. Regardless of its exact value, the existence of the limit
172:
Examination of available numerical data for known values of
301:
262:{\displaystyle \pi (x)\approx {\frac {x}{\log(x)-B}},}
838:
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428:
363:
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311:
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95:
Later elements up to 10,000,000 of the same sequence
29:
without a subscript base should be interpreted as a
415:{\displaystyle \pi (x)\sim {\frac {x}{\log(x)-B}},}
16:
Constant of proportionality of prime number density
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165:. The value that corresponds precisely to its
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832:Sur la distribution des zéros de la fonction
8:
64:The first 100,000 elements of the sequence
937:
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741:. New York: Springer-Verlag. p. 188.
204:Legendre constructed in 1808 the formula
201:led Legendre to an approximating formula.
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560:It is an immediate consequence of the
129:occurring in a formula constructed by
7:
776:"On Legendre's Prime Number Formula"
334:with a "very satisfying precision".
133:to approximate the behavior of the
660:
446:
14:
861:et ses conséquences arithmétiques
781:The American Mathematical Monthly
553:proved in 1849 that if the limit
513:provided that this limit exists.
305:), as giving an approximation of
739:The Little Book of Bigger Primes
723:Essai sur la théorie des nombres
337:Today, one defines the value of
1122:Conjectures about prime numbers
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689:Charles de La Vallée Poussin
675:(for some positive constant
422:which is solved by putting
169:is now known to be 1.
33:, also commonly written as
1158:
950:
854:{\displaystyle \zeta (s)}
737:Ribenboim, Paulo (2004).
288:{\displaystyle B=1.08366}
945:Prime number conjectures
726:. Courcier. p. 394.
720:Legendre, A.-M. (1808).
687:), as proved in 1899 by
1096:Schinzel's hypothesis H
327:{\displaystyle \pi (x)}
194:{\displaystyle \pi (x)}
158:{\displaystyle \pi (x)}
135:prime-counting function
1142:Analytic number theory
1127:Mathematical constants
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1101:Waring's prime number
901:"Legendre's constant"
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774:Pintz, Janos (1980).
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131:Adrien-Marie Legendre
127:mathematical constant
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562:prime number theorem
545:prime number theorem
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106:) −
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1066:Legendre's constant
167:asymptotic behavior
123:Legendre's constant
1017:Elliott–Halberstam
1002:Chinese hypothesis
898:Weisstein, Eric W.
870:2012-07-17 at the
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551:Pafnuty Chebyshev
536:{\displaystyle B}
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350:{\displaystyle B}
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31:natural logarithm
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890:External links
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788:(9): 733–735.
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762:Edmund Landau
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748:0-387-20169-6
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543:implies the
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1081:Oppermann's
1027:Gilbreath's
997:Bunyakovsky
683:(…) is the
102:= log(
71:= log(
1132:1 (number)
1116:Categories
1086:Polignac's
1059:Twin prime
1054:Legendre's
1042:Goldbach's
972:Agoh–Giuga
707:References
357:such that
1071:Lemoine's
1012:Dickson's
992:Brocard's
977:Andrica's
906:MathWorld
840:ζ
802:0002-9890
661:∞
658:→
634:
623:−
592:
574:π
481:π
472:−
460:
447:∞
444:→
401:−
389:
377:∼
365:π
313:π
248:−
236:
224:≈
212:π
180:π
144:π
1137:Integers
1076:Mersenne
1007:Cramér's
868:Archived
679:, where
651:as
1032:Grimm's
982:Artin's
810:2321863
691:, that
302:A228211
300::
283:1.08366
865:Online
808:
800:
745:
269:where
1091:PĂłlya
806:JSTOR
518:1.083
125:is a
1047:weak
798:ISSN
743:ISBN
298:OEIS
23:log(
965:2nd
960:1st
790:doi
631:log
457:log
437:lim
386:log
233:log
43:log
41:or
35:ln(
1118::
903:.
804:.
796:.
786:87
784:.
778:.
589:Li
547:.
520:66
938:e
931:t
924:v
909:.
849:)
846:s
843:(
812:.
792::
751:.
693:B
681:O
677:a
655:x
645:)
637:x
626:a
619:e
615:x
611:(
607:O
604:+
601:)
598:x
595:(
586:=
583:)
580:x
577:(
555:B
531:B
501:,
497:)
490:)
487:n
484:(
477:n
469:)
466:n
463:(
453:(
441:n
433:=
430:B
410:,
404:B
398:)
395:x
392:(
382:x
374:)
371:x
368:(
345:B
322:)
319:x
316:(
295:(
280:=
277:B
257:,
251:B
245:)
242:x
239:(
229:x
221:)
218:x
215:(
189:)
186:x
183:(
153:)
150:x
147:(
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108:n
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